Download magnetic ordering phenomena and dynamic fluctuations in cuprate

MAGNETIC ORDERING PHENOMENA AND
DYNAMIC FLUCTUATIONS IN CUPRATE
SUPERCONDUCTORS AND INSULATING
NICKELATES
HANS B. BROM1 and JAN ZAANEN2
1
Kamerlingh Onnes Laboratory, Leiden University, NL 2300 RA
2
Lorentz Institute for Theoretical Physics, Leiden University, NL 2300 RA
Handbook of Magnetic Materials, Vol. xx
Edited by K.H.J. Buschow
2003 Elsevier Science B.V. All rights reserved
1
CONTENTS
1. Introduction
2. Microscopic theories of stripe formation
2.1. What causes stripes?
2.2. The early work: mean-field theory
2.3. Improving on mean-field
2.4. Stripes, valence bonds and superconductivity
2.5. Considerations based on the t − Jz model
2.6. The ’exact’ numerical approaches
2.7. Frustrated phase separation
2.8 The role of the dynamical lattice
3. The magnetic resonance: theoretical aspects
3.1. The facts to be explained
3.2. Symmetry (I): spin modes and the coexistence phase
3.3. Symmetry (II): SO(5) and the π particles
3.4. Restoring the fermions: the various viewpoints
3.5. The resonance peak and spin-charge separation
3.6. The resonance peak and superconductivity
3.7. Concluding remarks
4. Experimental Techniques
4.1. NMR
4.1.1. The NMR method
4.1.2. Metals, superconductors and magnetic materials
2
4.1.2.a. Magnetic materials
4.1.2.b. Strongly correlated metals
4.1.2.c. Superconductors
4.2. µSR
4.3. Neutron scattering
4.4. ARPES
4.5. STM
4.6. ”Thermodynamic” techniques, like susceptibility and heat capacity
5. The hole-doped single-layer 214 cuprates
5.1. Neutron scattering
5.2. µSR and NMR/NQR
5.3. Results of other magnetic techniques
5.4. Charge sensitive techniques - X-ray, conductance and ARPES
5.5. Summary
6. Oxygen and strontium doped 214-nickelates
6.1. Phases and structural changes under oxygen and strontium doping
6.2. Neutron scattering
6.3. µSR, NMR/NQR and magnetic susceptibility
6.4. Charge sensitive techniques - X-ray conductance and ARPES
6.5. Summary
7. The electron-doped single-layer compound Nd2−x Cex CuO4+y
7.1. Charge sensitive data
7.2. Spin sensitive results
7.3. Summary
8. The hole-doped double-layer 123-compounds
8.1. Neutron scattering
8.2. µSR and NMR/NQR
8.3. Summary
3
9. Other multilayered cuprates - Bi-2212
9.1. Neutron scattering
9.2. µSR and NMR/NQR
9.3. Charge sensitive techniques
9.4. Summary
10. Concluding remarks
Acknowledgements
Appendix A. Static and dynamic properties of 2 dimensional Heisenberg antiferromagnets
References
4
1. Introduction
The rich phase diagram of the so-called high-Tc superconductors (Bednorz and Muller
1986), i.e. superconductors with surprisingly high superconducting transition temperatures
Tc , as function of doping strongly stimulated the interest in the interplay between antiferromagnetism and superconductivity. The parent compounds La2 CuO4 and YBa2 Cu3 O6 of the
high-Tc superconductors (HTC) La2−x Srx CuO4+δ and YBa2 Cu3 O6+δ , are two-dimensional
(2D) Heisenberg antiferromagnetic insulators. The spins reside mainly on the Cu ions and
are antiferromagnetically aligned along the CuO-axes. The two-dimensional nature arises
from the fact that the electronic spins responsible for the magnetism reside in the transition
metal oxide layers, see fig. 1, with a negligible coupling between the adjacent planes.
The electric insulating character of La2 CuO4 (or 214-compound) and YBa2 Cu3 O6 (or
123-compound) has a different origin from that of conventional insulators. In ordinary
insulating systems, the electric conductivity is hindered by the Pauli exclusion principle.
The motion of electrons is forbidden when the conduction band is filled, i.e. the contributing
atomic orbitals all donate two electrons. Strongly correlated electron systems, of which the
parent compounds of the HTCs are good examples, turn insulating due to a strong repulsion
between electrons and are known as Mott-Hubbard insulators (Gebhard 1997).
The insulating and antiferromagnetic properties of the 214 and 123 parent-compounds
disappear rapidly upon chemical doping. The doping is regarded as injecting holes or electrons into the two-dimensional antiferromagnetic layers. Holes are primarily located on the
p orbitals of the four oxygens surrounding the central copper in the CuO2 plane and compensate the copper spins in so-called Zhang-Rice singlets (Zhang and Rice 1988), see fig. 2.
In practice, hole doping is primarily achieved by either replacing some of the La3+ ions by
(2+)-ions ( Sr2+ or Ba2+ ), by adding excess oxygen to stoichiometric La2 CuO4 or by both
methods.
Fig. 3 gives the phase diagram as function of doping for the 214 and 123 cuprates.
The first phase is the antiferromagnetic region, bounded by the Néel ordering temperature
5
TN , and is theoretically well understood (Batlogg and Varma 2000). A summary of the
theory behind these Heisenberg antiferromagnets is given Johnston in the Handbook of
Magnetic Materials, vol. 10 (Johnston 1997), to which we refer to as HMM-10 (in the
appendix we briefly repeat the most important aspects). At higher doping levels the system
moves into a superconducting phase, for which at the optimal doping level xopt ≈ 0.15 the
maximum Tc is achieved. The optimal level separates the underdoped and overdoped regimes
respectively below and above xopt , see fig. 3. The 214 and 123 compounds can be doped
from 0 (antiferromagnetic phase) up to the overdoped regime. The pristine multilayered
compounds with Bi, Tl, or Hg are often already close to optimally doped. Especially in
the underdoped regime, subtle properties of the specific compound determine the particular
phase at zero temperature (T = 0 K).
The ground state at T = 0 K as function of the control parameter g and doping level x
is sketched in fig. 4, and is generic for hole-doped high-Tc superconducting compounds. The
control parameter g will depend on the spin stiffness and coupling to the lattice. In the 2212
compounds doping levels are typically above 0.15 and samples go from a superconducting
to a Fermi surface phase. The 214 and 123 compounds allow lower doping levels, where a
coexistence phase of stripes and superconductivity is possible. Stripes, see fig. 5, might be
described as rivers of charge in an antiferromagnetic background. Only in La2−x Srx CuO4+δ
the stripe phase exists in a large x regime. At low doping levels the magnetic phase remains
insulating. Spin and charge separation occurs as in the insulating nickelates (we might
speak of classical stripes). Higher doping leads to a conducting stripe and superconducting
coexistence phase, where the charges are more delocalized than in the nickelates. In the
heavily doped regime signatures of stripes disappear. By adding Nd the low temperature
orthorhombic structure of the 214-compounds is transformed to low temperature tetragonal
phase and the pinning of stripes is promoted, but superconductivity is suppressed.
HMM10 contains a thorough discussion of two-dimensional antiferromagnetism (briefly
summarized in the Appendix) and the properties of the following important two-dimensional
Cu-monolayer compounds Sr2 CuO2 Cl2 , La2 CuO4 , Ca0.85 Sr0.15 CuO2 , Nd2 CuO4 , the related
6
linear chain compound (Ca, Sr)2 CuO3 and spin ladder compounds. At the end also the
effects of phase separation and stripe formation after oxygen and Sr doping in La2 CuO4
are presented. Especially in this area progress has been considerable during the last couple
of years. We will update Johnston’s review by describing the fascinating phenomena in
hole doped La2 CuO4 (sections 5) and La2 NiO4 (section 6), and electron doped Nd2 CuO4
(section 7) regarding the stripe phases, which have appeared since 1997. We will also cover
the CuO-multilayer compounds (section 8 and 9), where the collective magnetic mode, the
so-called Resonance Peak (Rossat Mignod et al. 1991a and 1991b, Mook et al. 1993, Fong
et al. 1995, Bourges et al. 1996), is the most intriguing topic from a magnetic point of view.
Section 4 gives details of the experimental techniques, like ARPES (Angular Resolved Photo
Emission Spectroscopy), STM (scanning tunneling microscopy), µSR (muon spin rotation),
and especially NMR and NQR (resp. nuclear magnetic and quadrupole resonance) with
relevant examples; details about neutron scattering are to be found in HMM-10.
High-Tc superconductivity is an unsolved problem. The two theoretical sections 2 and 3
are dedicated to two aspects of the problem which have clearly to do with magnetism: the
static stripes and the magnetic resonance. These sections are not intended as an exhaustive
review of everything ‘magnetic’ of relevance to the problem of high-Tc superconductivity.
We have just focussed on the two aspects of the problem where we feel that in recent
years a sense of consensus has emerged. In section 3, we deliberately focus on what can
be called strong, static stripe order. In fact, the case can be made that the ‘strong’ stripe
orders as they occur in nickelates, manganites and so on are in essence understood. In the
cuprates this is less clear cut. In some regards, the static stripes in cuprates look quite like
nickelate, etcetera, stripes but in other regards they are still mysterious. This has to be
related to the fact they live in close proximity to the superconductors. Quantum effects
are supposedly far more important and one can wonder if these are really understood. The
next step is the idea of dynamical stripes. This corresponds with the notion that even in
the fully developed superconductors correlations of the stripe kind persist, but only in some
fluctuating form. In first instance, this notion should be viewed as an empiricism: some data
7
are just quite suggestive in this regard and due attention will be paid to dynamical stripes
in the experimental sections. However, for the theorists it is not an easy subject. Although
quite a number of interesting ideas have emerged it is at present not easy to present them in
a coherent fashion. It is the kind of subject matter which is more suited for monograph style
reviews and several of these are right now under construction. Two such pieces are available:
Zaanen et al. (2001), with an emphasis on the dynamical stripe issue, and Carlson et al.
(2002) with a primary focus on how superconductivity and stripes hang together. Both
reviews are largely complementary to what is presented here.
High-Tc superconductivity is also a highly controversial subject. Considering (dynamical)
stripes to be at one end of the controversy, the theoretical literature associated with the
magnetic resonance is clearly at the opposite end. Much theoretical work has been devoted to
the latter, and this will be reviewed in Section 3. The resonance is a dynamical phenomenon
which goes hand in hand with the superconductivity itself. While (dynamical) stripes are
understood as a highly organized form of electron matter, having no evident relationship
whatever with the Fermi-gas wisdoms of conventional metal physics, the resonance finds
a surprisingly detailed explanation in terms of this fermiology language. Although the
underlying interpretations can be quite different, the punch line is that the physics of the
resonance can be successfully traced back to the existence of a Fermi-surface, undergoing
a BCS instability into a d-wave superconductor. In fact, this section can also be read as a
tutorial introduction of the main-stream theories of high-Tc superconductivity. The notion
that the mechanism causing the superconductivity has to do with ‘something magnetic’ has
been very popular all along, like in the approaches where magnetic fluctuations behave like
phonons, or in the more exotic SO(5) and spin-charge separation theories. In this mindset,
the resonance appears as the ‘magnetic glue’ becoming visible in experiment.
We hope that by taking together the sections 2 and 3 the reader acquires some understanding of the depth of the high-Tc mystery. Pending on the focal point of the experimental machinery, one seems to enter completely different universes (stripes, fermiology)
which might appear even to be mutually exclusive. We have a strong suspicion that this
8
has little to do with the incompetence of experimentalists or with theorists being carried
away by their wild phantasies. Instead, we like to interpret it as the ‘theory of everything
in high-Tc superconductivity’ signalling to us that humanity has not quite figured out yet
the real nature of the thing.
Johnston (1997) lists relevant reviews about the magnetic and other properties of highTc superconductors, which have been presented before 1997. Of the new reviews that have
appeared since then, we mention the work of Imada, Fujimori and Tokura (1998) about
the metal-insulator transition. Starting from correlation effects, the authors describe the
various theoretical approaches to the unusual metallic states and to the metal-insulator
transition. The treatise contains a special section devoted to high-Tc cuprates, which includes
a discussion of YBa2 Cu3 O7−y and Bi2 Sr2 CaCu2 O8+δ . Regarding NMR and NQR on stripes,
the article of Hunt et al. (2001) might serve as a review as well. The reader likely will also
enjoy the very accessible articles of Goss Levi (1998), Brooks (1999), Emery, Kivelson, and
Tranquada (1999), Zaanen (1998 and 2000b), Laughlin, Pines, Schmalian, Stojkovic and
Wolynes (2000), Buchanan (2001), and Laughlin and Pines (2000).
9
FIGURES
10
FIG. 1. The CuO2 layers in the high-Tc compounds. The charges are mainly located on the
oxygen sites (open circles). The electronic spins responsible for the magnetism reside mainly on
the Cu ions (arrows) and are antiferromagnetically aligned along the CuO axes. In La 2 CuO4
the antiferromagnetic CuO2 planes are sandwiched between two non-magnetic La2 O2 layers. In
YBa2 Cu3 O6 , CuO2 double-layers share an yttrium site located in the center of the two Cu-squares.
As in the single-layered cuprates the double layers are separated by non-magnetic layers of Ba 2 O2
and a layer of Cu atoms (so-called Cu(1) sites, in contrast to the Cu(2) sites of the double layer). For
La2−x Srx CuO4 the dimensions for the enlarged orthorhombic unit cell are 0.536, 0.540 and 1.316
nm (x = 0), and 0.350, 0.535, and 1.321 nm for x = 0.1. Above x = 0.21 the structure is tetragonal
with a unit cell of 0.377 and 1.325 nm (HMM10). For YBa2 Cu3 O6+x the (quasi-)tetragonal unit
cell dimensions are 0.385 and 1.183 nm for x = 0 and 0.388, 0.383 and 1.163 nm for x ∼ 1, with a
distance between the two CuO2 layers of the double layer of 0.335 nm. These cell dimensions are T
dependent: for x = 0.8 the c-axis changes from 1.182 to 1.178 nm, while the a, b axes shrink from
0.3851 to 0.3847 nm (Rossat-Mignod 1991a, 1991b, 1992). For the Bi 2 Sr2 Can−1 Cun O2n+4 family
we only discuss the n = 2 compound Bi2 Sr2 CaCu2 O8 or Bi2212. The structure is simply related
to Bi2201, by replacing the CuO2 monolayer by a CuO2 /Ca/CuO2 sandwich. The structure is
pseudo-tetragonal and has a Tc of 85 K. Tc can be controlled by changing the oxygen concentration
or by substituting Y or other rare-earth elements for Ca. As grown samples are close to optimal
doping. Similar structures can be made based with Tl in stead of Bi. The c-axis of the Bi2212 or
Tl2212 compounds is 2.932 nm and for a triple layer, like Tl2223, 3.588 nm.
11
FIG. 2. Schematic diagram for the Zhang-Rice singlet that shows the hybridization of the
Cu and O orbitals, after Zhang and Rice (1988). The four oxygen-hole states can form either
a symmetric or antisymmetric state with respect to the central Cu-ion. The (+) and (−) signs
represent the phase of the p- and d-states wave-functions. Both symmetric and antisymmetric
states may combine with the d-wave Cu hole-state to form either singlet- or triplet-spin states.
The symmetric state forms bonding and antibonding states, where the large binding energy in the
singlet state is due to the phase coherence. The antisymmetric state is non-bonding. The energy
of two O holes residing on the same square is much higher than the energy of two separated O
holes: two holes feel a strong repulsion on the same square. Note that the localized states are
not orthogonal to each other, because the neighboring squares share a common O-site. Zhang and
Rice argue that the physics remains essentially unchanged if neighboring squares are taken into
account.
12
2.0
TN
1.0
0.5
0.0
antiferromagnetic
Temperature
1.5
0.0
optimally doped
underdoped
Tc
overdoped
super Conducting
conducting
0.2
0.4
0.6
doping level
0.8
1.0
FIG. 3. Phase diagram for high-Tc compounds showing the transition temperatures as function
of hole doping. In La2−x Srx CuO4+δ the doping level is due to doping by either Sr2+ , excess
oxygen or both and is quantified as x + 2δ. The symbol Tc refers to the superconducting transition
temperature. TN is the boundary for the Néel-ordered antiferromagnetic phase. The 214 and
123-compounds can be doped from 0 (antiferromagnetic phase) up to the overdoped regime. The
doping levels of the multilayered compounds with Bi, Tl, or Hg are often in the optimally doped
and overdoped regime, while strongly underdoped values are hard to realize. Especially in the
underdoped regime, subtle properties of the specific compound determine the particular phase at
T = 0 K, see also fig. 4.
13
Fermi
Surface
Phase
1/g
stripe phase
classical
stripes
coexistence
phase
214(Nd)
214
d-wave
superconductor
0.0
0.1
x
123
2212
0.2
0.3
FIG. 4. Generic phase diagram at T = 0 K for hole-doped high-Tc superconducting compounds showing the ground state as function of a control parameter g (which will depend on the
spin stiffness and coupling to the lattice) and doping level x. In the 2212 compounds doping levels
are typically above 0.2 and samples go from a superconducting to a Fermi surface phase. The
123-compounds allow lower doping levels, where a coexistence phase of stripes and superconductivity is possible. Only in La2−x Srx CuO4+δ the stripe phase exists in a large x regime. At low
doping levels the magnetic phase remains insulating. Spin and charge separation occurs as in the
insulating nickelates (one might speak from classical stripes as the quantum character of strong
stripe fluctuations has become less important). Higher doping leads to a conducting stripe and
superconducting coexistence phase, where the charges are more delocalized than in the nickelates.
In the heavily doped regime signatures of stripes disappear. By adding Nd the low-temperature
orthorhombic structure of the 214-compounds is transformed to low-temperature tetragonal and
the pinning of stripes is promoted, while superconductivity is suppressed.
14
15
FIG. 5. Possible stripe structures in the hole-doped 2-dimensional CuO2 layer. (Upper panel)
Definition unit cell and directions. Open circles represent oxygen sites and filled circles 3d-ions.
The unit cell in the CuO2 plane is formed by connecting the four nearest neighbor oxygens surrounding the Cu-ion (the Zhang-Rice configuration, see fig. 2). The drawn square on the right
is the enlarged unit cell. (H, V) refers to the orientation of a horizontal or vertical domain wall.
B refers to a bond-centered, D to a site-centered diagonal domain wall. (Lower panel) Schematic
diagrams for an ideal diagonal static-striped-phase ground-state at 1/3 hole doping, as suggested
theoretically by Zaanen and Gunnarson (1989) and observed later experimentally for hole-doped
La2 NiO4 (Tranquada et al. 1995 and 1997b). The charged domain walls can be either site-centered
(left) or bond-centered (right). In the case of diagonal stripes with hole doping of 1/3 for site centered domains the domains are two spins wide, and the spins within the domain are equivalent and
coupled by exchange interaction (J), while spins on both sides of the stripe are aligned antiparallel
with exchange J 0 < J. For oxygen or bond centered stripes, the domains are 3 spins wide, and an
uncompensated moment might appear. The magnetic moments on sites adjacent to a charge stripe
are likely to be reduced in magnitude compared to those in the middle of the domain. Perfect
compensation is not expected. In both cases the phase of the antiferromagnetic order shifts by
π on crossing the domain wall, hence for a bond-centered stripes spins adjacent to the stripe are
ferromagnetically aligned.
16
2. Microscopic theories of stripe formation.
To avoid ambiguities, stripes should be in first instance associated with the highly organized form of electron matter as encountered in for instance the nickelates (section 6) but
also the manganites. These stripes should be appreciated as a generic order associated with
the doped Mott-insulator, in the same sense that antiferromagnetism belongs to the Mottinsulating state itself. As it turns out, the cartoon picture of ‘rivers of charge’ separated by
Mott-insulating magnetic domains is surprisingly literal in these systems.
Fully developed stripe order of this kind goes hand in hand with a strong insulating
electrical behavior, and as such they are quite opposite to superconductors. The stripes in
the cuprates, even the static ones, are a bit special. They live in close proximity, or even
coexist with superconductivity, which has to imply that the stripe order as found in e.g. the
nickelates is to a further or lesser extent compromised in the ‘stripy’ cuprates. In this regard
we strongly recommend the newcomers to this field to have a close look at the nickelate
stripe systems first, because they offer a simple setting to get used to the basic physics.
It all starts out with the Mott-insulator itself, which is quite different from the band
insulator. Even the problem of a single hole in the Mott-insulator is non-trivial, see Dagotto
(1994) and ref’s therein. When the hole moves through the dynamical spin system it causes
severe spin frustrations, which in turn frustrates the free motion of the hole. In fact, the
single hole can only propagate as a quasiparticle due to the presence of quantum spin fluctuations repairing the damage in the spin background. However, when the density of holes
is finite they can help each other repairing the spin damage by concerting their motions.
This is in simple words the origin of the complex ordering phenomena called stripes.
2.1. What causes stripes?
Stripes were discovered theoretically shortly after the discovery of superconductivity and
long before they were seen experimentally, Zaanen and Gunnarsson (1989) (this paper was
17
delayed for ∼ 1.5 years by the refereeing process !). There is no such thing as a full proof
theoretical method in this branch of science and one always relies on theoretical toy models
which are treated in a more or less uncontrollable approximation. The original prediction
was based on the standard Hubbard toy-models, treated in a most conventional fashion:
Hartree-Fock electronic mean-field theory. We will discuss this at some length in section
2.3 because it is conceptually interesting, while Hartree-Fock is actually quite successful in
the description of the stripes of the nickelate variety. Subsequently, it became clear that
various approximate methods designed to improve quantitatively on Hartree-Fock share the
generic tendency towards stripe formation (section 2.3). Implicitly, Hartree-Fock and its
descendants are controlled by large spin: it is designed to describe ‘classical’ antiferromagnetic order, and the quantum fluctuations in the spin system expected to be important for
S = 1/2 are just neglected. Instead one can start in the limit dominated by these fluctuations and this is theoretically achieved by generalizing the models to SU (N ) (or Sp(2N ))
symmetry, to consider the limit that N is large while S is small. Sachdev and coworkers
(1999) discovered that also in this limit stripe like ordering phenomena occur, going hand
in hand with superconductivity, and these will be discussed in section 2.4. In section 2.5
we will turn attention to the various attempts to acquire further insights using the holeand spin language of the t − J model, to continue in section 2.6 with a discussion of the
numerical ‘exact’ results with an emphasis on the DMRG (density matrix renormalization
group) work by White and Scalapino (1998). In the remaining sections we will focus on the
bits and pieces of the physics which are ignored in the Hubbard- and t−J model for no good
reasons: long-range Coulomb-interaction (section 2.7) and the electron-phonon interaction
(section 3.8). Although initially perceived as offering competing explanations for stripe formation, more recently a consensus has emerged that these effects add up to the Hubbard
model physics, to cause stripes to be even more ubiquitous.
2.2. The early work: mean-field theory.
18
The idea that inhomogeneity could be intrinsic to strongly correlated electron systems
has been around for some time, particularly in the Russian school (Kateev et al. 1998).
However, these ideas were rather handwaving and stripes came into existence in the late
1980’s, when Zaanen and Gunnarsson (1989) discovered the stripes in the mean-field theory
of Hubbard models. It was directly realized that these ‘charged antiferromagnetic domain
walls’ should be viewed as close relatives of the Su-Schrieffer-Heeger type solitons (and
holons) of one dimensional physics (Zaanen and Gunnarsson 1989). The simple mean-field
calculations got this novel and crucial aspect of ‘topological doping’ right, following the
terminology introduced by Kivelson and Emery (1998).
In hindsight this is not too surprising because Hartree-Fock is controlled by a powerful
principle. It is often misinterpreted in the quantum-chemist’ fashion, as a not so good
quantitative approximation of the interaction term in a small system Hamiltonian. In physics
it has a much stronger meaning: it is the theory telling how electrons collectively manage to
cause ‘conventional’ order, like spin density waves (antiferromagnets), charge density waves
(Wigner crystals) and BCS superconductors.
As an example, let us consider the well-known Hubbard hamiltonian
Ĥ = −t
X
(a†iσ ajσ + a†jσ aiσ ) + U
X
ni↑ ni↓ ,
(1)
i
<ij>σ
describing a system of tight-binding fermions (creation and annihilation operators a †i resp.
ai , spin label σ, and occupation number operator ni , all for site i) with a transfer integral
t, subjected to a local repulsion ∼ U . Except for the one- and infinite dimensional cases,
this Hamiltonian cannot be solved exactly. However, in the case that electron systems break
a symmetry of eq. (1) spontaneously, vacuum expectation values develop; broken (charge)
translational symmetry means hρi i = hni↑ + ni↓ i 6= 0 and broken spin rotational invariance
2hSiz i = hni↑ − ni↓ i 6= 0. Given that the order exists, the number operators appearing
in the full hamiltonian turn into scalars, implying that the interaction term turns into a
single particle self-consistent potential, thereby simplifying the problem to an extent that it
becomes generally solvable. Write niσ = hniσ i + δniσ , and it follows that,
19
U
X
i
ni↑ ni↓ = U
X
i
(hni↑ ini↓ + hni↓ ini↑ − hni↑ ihni↓ i + Hf luct
(2)
This is just a rewriting of the interaction term, but the crucial observation is that H f luct =
U
P
i
δni↑ δni↓ describes the fluctuations around the ordered state. The implication is that in
the presence of order, the mean-field hamiltonian obtained by neglecting Hf luct is a zeroth
order Hamiltonian having the property that its ground state can be adiabatically continued
to the true ground state: the perturbation theory in terms of Hf luct is converging. Therefore,
mean-field theory rests on the presence of order. According to a popular myth it would only
be good for small interaction strength, U/t << 1. This is nonsense. One can easily show
that at half-filling and for large U one recovers the magnetic insulator: just compare the
total energies of the antiferromagnet and the ferromagnet to find that they differ by the
superexchange J = t2 /U , while the paramagnetic metal has a very high energy, ∼ U/2.
Schrieffer, Wen and Zhang (1989) demonstrated that one recovers linear spin-wave theory
when Hf luct. is evaluated on the Gaussian level. In fact, Hartree-Fock is controlled by the
‘smallness of h̄’, translating into the largeness of spin (S → ∞) in the Mott-insulators, as
becomes particularly obvious in the field-theoretic (path-integral) treatment, see Fradkin
(1989).
Energy is minimized in the half-filled system by the uniform antiferromagnet. However,
there is no obvious reason for this to be the case in the doped systems. It is actually so
that the uniform mean-field solutions of the doped state are characterized by a negative
compressibility indicating that these are unstable. This is not only true for the collinear
magnets but also for the spin spiral states (see Dombre 1990 and Stojkovic et al. 1999) as
suggested originally by Shraiman and Siggia (1989) which are also based on Hartree-Fock.
As first realized by Schrieffer et al. (1988a, 1989), even for the simple case of a single hole the
mean-field ground state is rather non-trivial: a self-trapped solution is found called the ‘spin
bag’. Schrieffer’s conjecture was that the spin bag would give rise to a pairing mechanism,
along the same lines as the bipolaron mechanism: two holes together can dig a deeper spin
‘hole’ than two holes separately, and this acts like an attractive pairing force. However, as
20
was discovered by Zaanen and Gunnarsson (1989), energy is further minimized by forming
many particle bound states: the stripes, see fig. 6. These consist of linear Mott-insulating
domains characterized by collinear Néel spin order, separated by lines of holes which are at
the same time Ising type domain boundaries in the antiferromagnet: the staggered order
parameter changes sign every time a charge stripe is crossed. Finally, every domain wall
unit cell contains a net single hole. These are quite literally like the stripes of the nickelates
(see section 6).
Where is this coming from within this mean-field ‘paradigm’ ? The key is that these
solutions are topological (the domain wall property) and these stripes can be viewed as just
two-dimensional versions of the well-known solitons associated with the semi-classical sector
of one-dimensional physics, as realized in electron-phonon systems, see Heeger et al. (1988).
As Mukhin (2000) and Matveenko and Mukhin (2000) demonstrate, in the one-dimensional
‘large-S limit’ Hubbard-chains, the process of soliton formation is in essence the same as in
the electron-phonon models. All one needs is a ‘parent’ system which is insulating because
of the potentials associated with a staggered order parameter: the Néel order parameter
plays a role equivalent to the dimerization wave in polyacetylene. Matveenko and Mukhin
(2000) consider an ansatz for the staggered order parameter mz = (−1)i hSiz i ∼ tanh(k0 x)
describing a kink or soliton such that the order parameter is pointing in the opposite z
directions away from the origin while it is vanishing at the origin (k 0 determines the width),
see fig. 7. Diagonalizing the (linearized) mean-field Hamiltonian for this potential yields a
state centered at the middle of the gap. At half-filling this state is occupied by one electron,
and it carries a net spin 1/2 quantum number (localized ‘spinon’). This is a relatively costly
excitation. However, by adding one hole this near mid-gap state becomes unoccupied and
the soliton becomes the ground state: the cheapest way to remove the hole is by taking it out
of a state as close as possible to the middle of the gap, while at the same time a soliton profile
costs less order parameter gradient energy than a polaronic/‘spin-bag’ profile. This state
carries only a charge e quantum and no spin: the localized ‘holon’. Subsequently, Matveenko
and Mukhin (2000) demonstrate that close to half-filling a periodic lattice is formed of these
21
solitons, while at high doping and weak interaction strength a phase transition follows to a
single harmonic Peierls wave. In summary, the mean-field ground states of one dimensional
doped Mott insulator are soliton lattices which can be viewed as the 1D version of the
ordered stripes.
The 2D case is mathematically more complicated and no closed analytical solutions
are available. However, by inspection of the numerical results the qualitative picture was
already obtained in the early Zaanen-Gunnarsson paper. The one dimensional solutions
tell that much energy can be gained by binding the holes to the magnetic domain ‘walls’.
However, there is clearly no space for the 1D ‘holons’ in the 2D space, because these would
cause infinitely long strings of spin mismatches. However, this spin frustration is avoided
by putting the ‘holons on a row’, thereby forming stripes. In terms of the electronic states,
this means that the motion of the electrons perpendicular to the stripes causes mid-gap
states (see Zaanen and Oleś 1996). However, the electrons can now also propagate along the
stripes, and these mid-gap states hybridize in a ‘mid-gap band’: fig. 8. The width of this
band is less than the Mott gap and by having a density of one hole per domain wall unit
cell, this mid gap band is empty. Hence, the thermodynamic potential is in the gap between
the occupied lower Hubbard band and the empty mid-gap band and this new band gap is
responsible for the stability of the stripe phase.
These early papers triggered a number of follow ups, investigating these matters in
further detail (Kato et al. 1990, Schulz 1989 and 1990, Poilblanc and Rice 1989, Inui and
Littlewood 1991, Verges et al. 1991). Altogether, these studies made clear that the meanfield stripes are quite robust and uniquely characterized by a number of properties: (a)
their topological nature is at the heart of the phenomenon, (b) starting from the standard
Hubbard-type problems, mean-field stripe phases are insulating, (c) this insulating character
goes in hand with a charge quantization rule stating that one hole stabilizes one domain
wall unit cell (‘empty’ stripes), (d) it costs little energy to bend stripes, to reorient them
from horizontal/vertical- to diagonal directions, etcetera. Of special interest is the work by
Viertio and Rice (1994), showing that the kinetic energy of kink motion can easily overcome
22
the cost of creating a kink. Hence, it appears that ‘meandering’ fluctuations of stripes can
be quite important for both the thermal- and quantum melting and this inspired studies on
the nature of the dynamical stripes (e.g., Zaanen et al., 2001).
Theory preceded the experimental fact and the above was quite instrumental in guiding
Tranquada and coworkers (Tranquada et al. 1995) in the process which lead to the discovery
of stripes in nickelates (section 6). However, nature came up with a surprise when the same
group discovered the stripes in the La2−x−y Srx Ndy CuO4 system: at doping levels x < 1/8
charge appears quantized but now 1 hole is associated with two domain wall unit cells, while
for x > 1/8 charge quantization seems to disappear altogether. This is often quoted as
proving that mean-field theory falls short fundamentally for the description of the cuprate
stripes, and this is a too strong statement. As Zaanen and Oleś (1996) showed, HartreeFock solutions exist corresponding with half-filled stripes (see also Bosch, van Saarloos and
Zaanen 2001). Although ‘empty’ stripes have a lower energy, these half-filled stripes are
locally stable and they have quite low energies: one could argue that the problem is not so
much in the approximation, but more in the choice of effective theory. More importantly, a
commensurate ‘internal’ (on-stripe) charge density wave is required for stability, causing a
gap in the spectrum of mid-gap states, see fig. 9. Hence, according to mean-field also the
half-filled stripes should be insulating which seems at first sight in contradiction with the
metal-like transport found in the striped cuprates, although it offers a natural explanation
for the on-stripe charge quantization seen in the regime x < 1/8. Attempts to explain this
with metallic stripes are not quite convincing (e.g. Nayak and Wilczek 1997).
2.3. Improving on mean-field.
As we already discussed, straightforward mean-field theory has in first instance a qualitative use. To become quantitative one has to evaluate the fluctuations with perturbation
theory and this is a difficult task. These fluctuations come in two varieties: (a) the longwavelength fluctuations which refer to the highly collective motions associated with the
23
order itself, like spin fluctuations/magnons but also the stripe meanderings, (b) the local
fluctuations which have to do with the problem that Hartree-Fock cannot be considered to
be a faithful representation of the electron dynamics on the lattice scale. Eventually, the
faith of stripes is decided on the microscopic scale and to arrive at meaningful quantitative
statements one should go beyond Hartree-Fock. Several methods emerged to improve the
short distance aspects. In order of increasing sophistication, these are the Gutzwiller meanfield and the closely related local-ansatz, the Gutzwiller Ansatz and the modern dynamical
mean field-theory (DMFT). These methods have all been tried out on the stripe problem,
confirming that stripes are quite robust.
The intrinsic weakness of Hartree-Fock is that it rests on a single Slater determinant
description of the electronic wavefunction. The central ingredient of Hubbard type problems
is that for large U the Hilbert space associated with the low energy dynamics is projected:
configurations containing doubly occupied sites are at high energy. The only way to recover
the correct Hilbert space within the limitations of a single determinant description is by
having a large order parameter because the unwanted states are automatically projected
out by the order parameter potential. Instead, one can attempt to directly attack this
problem by brute force using the configuration-interaction approach of quantum chemistry:
see Louis et al. (2001) for an interesting recent attempt in the stripe context.
In classical statistical physics (and bosonic quantum problems) mean-field theory becomes always exact in the limit of infinite space dimensions (D → ∞). Based on earlier
work by Metzner et al. (1989), Georges et al. (1992) discovered that in infinite dimensions
fermionic problems simplify. However, their ‘time axis stays active’ and they are described
in terms of an interacting impurity problem (Kondo-type problems) coupled self-consistently
to a fermionic bath which is itself constructed from the solutions of the single impurity problem. In the dynamical mean-field theory one uses the exact results of infinite dimensions
to calculate matters in finite dimensions: it amounts to an excellent treatment of the local
physics while it inherits from its D → ∞ origin the complete neglect of the long wavelength
fluctuations. The Gutzwiller Ansatz, as well as the closely related local-ansatz can be con24
sidered as approximations to the DMFT. For a recent review we refer to Georges et al.
(1996).
The first application of this methodology to the stripe problem is the work by Giamarchi
and Lhuillier (1990) employing the Gutzwiller Ansatz. For recent extensions we refer to
Yanagisawa et al. (2002), Seibold et al. (1998), Sadori and Grilli (2000), as well as Gora
et al. (1999) for work based on the local Ansatz. The highlight is the work by Fleck et al.
(2000, 2001) solving the DMFT equations for a large Hubbard model supercell, allowing full
variational freedom with regard to the charge and spin ordering. It suggests a picture which
is in some regards different from the Hartree-Fock outcomes. First, the doping dependence
seems identical to the experiments: for x < 0.05 empty diagonal stripes are most stable,
switching to half-filled horizontal stripes for x > 0.5 which extend up to x = 1/8. In the
doping range 0.125 < x < 0.20 the stripe ordering wave vector does not change any longer,
indicating that the charge density inside the stripe is no longer commensurate. Finally,
stripes disappear in the overdoped regime, x > 0.20. Secondly, these DMFT stripes are
charge-compressible: no energy gap is found in the spectrum and there is no tendency to
form internal density waves on the stripes. Thirdly, Fleck et al. (2000, 2001) show that
the DMFT self-energy shuffles the mean-field spectrum considerably, and the result is a one
electron spectral function which starts to resemble the experimental results. In summary,
the DMFT calculations offer a striking similarity to experiment although it is not always
clear where it is coming from.
2.4. Stripes, valence bonds and superconductivity.
Are stripes good for, or even causing the superconductivity, or is it that the stripes are
hindering the superconductivity? At long wavelength, stripe and superconducting order do
always compete in a restricted sense. As explained in section 3.2, they can coexist in the
sense that the same electrons divide their time between the stripes and the superconductivity.
However, stripe order also means charge order and this can only happen at the expense of
25
the superconductor. Superfluid density is at maximum when all bosons can move freely and
by partly crystallizing the bosons one takes away part of these free motions. Another issue
is, however, could it be that stripes help the formation of these bosons, the Cooper pairs?
Several theoretical groups came up with tentative answers to this question and these
can be summarized as: stripes are a way to organize the resonating valence bond (RVB)
mechanism of high-Tc superconductivity by Anderson (1987). Anderson conjectured that
the natural quantum competitor of the antiferromagnet is a state which is constructed from
bound pairs of S = 1/2 spins: the valence bonds or spin-dimers. One can cover the lattice
with many configurations of these dimers and he envisaged that due to strong quantum
fluctuations these dimers would form a featureless quantum fluid: the ‘resonating valence
bond state’. Upon doping with holes, the spins turn into electrons which can move around,
but because their spins are bound in pairs, the electrons are bound in Cooper pairs and the
doped state is therefore a superconductor characterized by a large pairing energy ∼ J.
What is the relationship with stripes? This relationship has been at the center of the
thinking of Kivelson and Emery and coworkers, and we refer to their review for a detailed
exposition, see Carlson et al. (2002). Let us just discuss the idea of the the ‘spin-gap
proximity effect’, Emery, Kivelson and Zachar (1997). The charge stripes are assumed to
be charge compressibility, and to zeroth order their internal electron systems are viewed
as independent one dimensional Luttinger liquids. The next assumption is that the spin
system living on the magnetic domains is a quantum paramagnet: the ground state is a
singlet, separated by an absolute energy gap from the lowest triplet excitations. These two
systems are not completely disconnected because the electrons confined in the charge stripes
can tunnel in and out the Mott-insulating domains. The tunneling probability will be higher
for pairs of Luttinger liquid electrons, because these can enter the domains as pair singlets
which directly ‘fit’ into the spin vacuum, while single electrons have to overcome the spin
gap energy because they carry S = 1/2. Hence, since electron singlet pairs move more freely
than single electrons, there is a pairing force at work of a kinetic origin, following the spirit
of the RVB idea.
26
Vojta and Sachdev (1999) and Vojta, Zhang, and Sachdev (2000) arrived at yet another
‘RVB-ish’ viewpoint. Although based on a limit (‘large-N ’) which is not necessarily of
physical relevance, it puts a strong emphasis on the quantum physics of the spin system.
Let us review in some detail this perspective which should be regarded as complementary
to the large-S limit. How to approach the limit where the quantum fluctuations of the
antiferromagnet are as important as can be imagined? One option is to increase the SU (2)
spin symmetry to a larger symmetry group. Keeping the ‘spin’ magnitude S small, the
larger symmetry will cause stronger fluctuations. For instance, one can consider SU (N ) or
the Sp(2N ) generalizations of the Heisenberg model, both reducing to SU (2) when N = 2.
It was discovered in the 1970’s in high energy physics that in the limit N → ∞, keeping
S finite, saddle points arise of quite a different nature than the large-S saddlepoints. This
notion became later popular also in condensed matter physics, playing a prominent role both
in heavy fermion physics, quantum magnetism and high-Tc superconductivity, see Auerbach
(1994).
In the large-N limit, at the moment that holes are introduced in the spin system superconductivity emerges. How does this work? The starting point is the well known t − J
model, for present purposes extended with a nearest-neighbor repulsion,
Ht−J−V = J
X
<ij>
~i · S
~j + t
S
X
<ij>
c̄†iσ c̄jσ + V
X
ni nj
(3)
<ij>
describing a system of electrons on a square lattice, under the constraint that there is either
a hole or an electron present on every lattice site, while two electrons are forbidden to occupy
the same site. The spins of neighboring electrons interact via a Heisenberg exchange (first
term) with strength J, while their charges interact via the (non-standard) repulsion V . In
the presence of holes, these electrons can delocalize as described by the t term, under the
constraint that no double occupancy occurs: the c̄† is not a normal fermion creation operator
but takes care of this projection which amounts to a great complication.
The large-N limit is now constructed as follows. The spins are described by fermions
fα† , α = 1, 2N , transforming under the fundamental of Sp(2N ). The holes are described by
27
spinless bosons b such that the Sp(2N ) ‘electron’ annihilation operator becomes c αi = fiα b†i
and the local constraint of the t − J model becomes
† α
fiα
fi + b†i bi = N
(4)
This is the usual operation in the ‘slave’ theories, corresponding with the mathematical
machinery underlying the idea of spin-charge separation (see also section 3.5). The constraint eq. (4) acts locally and this can be enforced by adding a Lagrange multiplier term
to the action ∼
P
i
³
´
† α
λi fiα
fi + b†i bi − N . Such a constraint is very hard to handle at finite
N where it requires the introduction of U (1) gauge fields. However, in the limit N → ∞
the local constraint becomes global λi → λ. When this condition is fulfilled, it becomes
straightforward to construct mean-field theories. The ‘holons’ (b’s) and spinons (f † ’s) become literal hard-core bosons and fermions. Nothing can prevent the bosons to condense
(except crystallization) such that hbi i 6= 0. Since the bosons carry electrical charge this is
a Meissner phase characterized by a large energy scale ∼ xt (x is doping). Hence, high-T c
superconductivity arises naturally and since the original proposal by Baskaran, Zhou and
Anderson (1987) this has attracted much attention. The standard view on these matters is
further discussed in section 3.5.
Besides the charge Bose-condensation, the spin system as encoded in the fermions
† †
fjσ0 i 6= 0. The standard
(‘spinons’) tends to BCS type (d-wave) pairing instabilities, hfiσ
mean-field phase diagram as discussed in section 3.5 is derived assuming spatial uniformity
of both the holon superfluidity and this spinon d-wave pair condensate. This uniformity
assumption is challenged by Vojta et al. (1999): in the N → ∞ limit spatially non-uniform
states have a lower energy and these states are quite similar to the stripes of the large-S
limit!
One should first get the state of the half-filled insulator right. As Read and Sachdev
(1989, 1990) showed, in the large-N limit energy is gained when the pair amplitudes break
in addition translational invariance, relative to the uniform d-wave BCS-like states. These
can be viewed as higher dimensional realizations of the spin-Peierls order known from one
28
dimensional physics. Sp(2N ) (or SU (N )) singlets are formed from nearest-neighbor spins
and these ‘dimers’ or ‘valence bonds’ are stacked in columns covering the lattice, see fig. 10.
Symmetry-wise, it is indistinguishable from a bond centered charge density wave, characterized by an energy (singlet-triplet) gap because it is commensurate with the lattice.
Alternatively, it can be viewed as a ‘frozen out’ form of Anderson’s RVB state.
It appears that these spin-Peierls orders are not unique to the large-N limit. Sachdev and
coworkers (Read and Sachdev 1990, and especially Sachdev and Park 2002) demonstrate that
these also merge starting from the large-S side. This is an elegant mechanism, rooted in the
Berry phase. Although the Berry phase can be ignored in ordered co-linear antiferromagnets,
it becomes active when the spins disorder quantum-mechanically. The ‘dynamics’ of the
Berry phase maps in this case on an effective electrodynamics which can be shown to describe
spin-Peierls order. These considerations suggest that the spin-Peierls state is a generic
competitor of collinear antiferromagnetic order, and it should therefore be also of relevance
to frustrated quantum-antiferromagnets. Although still controversial, in several numerical
studies on J1 − J2 Heisenberg models evidences are reported in favor of spin-Peierls order
(du Croo de Jongh, van Leeuwen and van Saarloos 2000 and Sushkov, Oitmaa and Weihong
2001a).
What happens when such a spin-Peierls spin state is doped? Qualitatively, the situation
is similar to the large-S case. The background state is protected by a commensuration gap,
and since the holes tend to drive the system away from commensuration, these are expelled.
The magnetic domains are now ‘built’ from the dimers, and since these wants to stack in
columns, stripes result according to the calculations by Vojta et al. (1999), see fig. 10. As
compared to the large-S stripes there are still important differences. First, the high doping
state which is locally realized inside the charge stripes corresponds now with a d-wave like
superconductor. This can be viewed as a superposition of a pair of holes and a spin dimer,
∼ U +V fi† fj† bj bi . This intra-stripe superconductivity turns into an overall superconductivity
due to inter-stripe Josephson coupling and one obtains an overall state which can be called a
supersolid: it is a coexistence state of (charge) crystalline order ( spin-Peierls and the stripe
29
density modulation) and superconductivity (with s + d symmetry because the system is
effectively orthorhombic). A second obvious difference is that these large-N stripes ‘live on
the links’: the charge stripes are bond centered, while the insulating domains prefer a width
corresponding with an even number of sites in order to maintain the spin-Peierls registry.
As Vojta et al. (1999) show, these ingredients add up to produce at least in an average
sense an incommensurability which is proportional to the hole density in the underdoped
(x < 1/8) regime, saturating at higher dopings caused by the diminishing strength of the
stripe order.
Summarizing, the large-N theory incorporates in a natural way superconductivity although it is not easy to account for the antiferromagnetism and the topological order. Instead, it rests on the spin-Peierls order which has never been seen in experiments. The
superconductivity originates in the strong quantum spin-fluctuations, being neglected in
Hartree-Fock. Martin et al. (2001) showed recently that it is possible to stabilize stripesuperconductivity coexistence states in Hartree-Fock. However, such states are only realized
in the presence of very large nearest-neighbor attractions. Nature is somewhere in the middle
and it is quite significant that the large-N and large-S stripes are quite similar: interpolating
between the two does not require much imagination.
Not so long ago, spin-charge separation and stripes were seen as opposing, mutually
exclusive viewpoints, but in the light of the latest results this attitude can no longer be
maintained. It makes clear that the tendency to form inhomogeneous structures supersedes
even the differences between insulators and superconductors (charge sector) or antiferromagnets and valence bond phases (spin sector). The theoreticians should be aware that their
preference for homogeneous states is in first instance based on the ease of computation, and
not on universal principle. The latest example might well be the flux phases as advocated
by for instance Wen and Lee (1996) and the closely related d-density wave order suggested
by Chakravarty et al. (2000). According to Chernyshyov and Wilczek (2000), stripes should
also be formed upon doping an insulator of this kind and preliminary numerical results by
Marston and Sudbö (2002, unpublished) suggest that such a kind of phenomenon indeed
30
occurs in their flux phase ladder systems, Marston, Fjaerestad, and Sudbö (2002). Finally,
it even seems that spins are not necessary for electronic stripe formation: Hotta, Feiguin
and Dagotto (2001) showed recently that the stripes in manganites have likely more to do
with orbital order than with spin order.
2.5. Considerations based on the t − Jz model.
Up to this point we have reviewed the outcomes of various mean-field approaches. These
theories are always controlled by limits and because nature is in the middle they are not
quite trustworthy. Is it possible to see where stripes are coming from, without invoking
mean-fields, i.e. in terms of the real motions of holes and spins? This is a very complicated
task and only computer calculations (the subject of the next section) can be conclusive.
Nevertheless, one can obtain quite some qualitative insights by considering simplified cases.
A common denominator in these attempts is to consider in first instance Ising spins instead
of Heisenberg spins. This simplifies matters considerably and the XY terms can subsequently
be considered as quantum ‘corrections’ changing the situation not too drastically.
The most important contribution in this regard is the early one due to Prelovs̆ek and
Zotos (1993). Their considerations were motivated by a numerical study of a small t − J
cluster, where they were the first to find indications for stripe correlations in the context
of the t − J model. They proposed a simple picture for stripe formation in terms of holes
moving through the Ising spin system, followed by a large scale numerical study suggesting
that the basic mechanism is of relevance for the full model. In its bare form, this picture
should be taken as a cartoon which is nevertheless most useful if it is handled with care.
Let us first consider what happens when a single hole hops through an Ising antiferromagnet; the Hamiltonian is just the t − J model, eq. (3) omitting the XY terms in the spin
part (‘t − Jz model’). The outcomes are well known (see Dagotto 1994) one direction, a spin
moves in the opposite direction and after a couple of hops one finds that the hole has left
behind a string of misaligned spins, see fig. 11a,b. For every line element of this ‘magnetic
31
string’ an energy penalty ∼ 4Jz has to be paid and this implies confinement: the hole cannot
move far away from its origin, costing a considerable kinetic energy. It is believed that this
magnetic string effect survives the quantum spin fluctuations: the XY terms ‘destroy’ the
string by flipping back the spins, thereby allowing the hole to propagate. However, this
implies that the kinetic energy scale ∼ J instead of the ∼ t of the free particle, as confirmed
by numerical simulations of the one hole problem.
Hence, the motions of an isolated hole are frustrated by the antiferromagnetic spin system. Prelovs̆ek and Zotos (1993) suggested that at finite densities the holes can ‘help each
other’ to avoid these frustrations. The recipe is simple: take as a starting configuration a
domain wall occupied by holes, in other words a stripe. When a hole moves transversally
to the stripe, the spin moves backwards meeting an antiferromagnetic-neighbor (fig. 11c).
Move the neighboring hole, and one sees two kinks appearing which propagate freely (∼ t)
when these processes are repeated (fig. 11d). These kinks have in turn the effect of changing
the position of the stripe in space and if they are freely created and annihilated the stripe
as a whole will delocalize over the plane. The conclusion is that although the motions of a
single hole are impeded by the spins, the stripe as a whole can meander rather freely. This
means a gain in kinetic energy, although a price has to be paid because every hole has to
‘watch’ its neighbors.
One can view it in yet another way: stripes are ‘holons on a row’, but now interpreted in
a more literal fashion as compared to the mean-field case (section 2.2). In one dimensional
physics spin-charge separation has a precise meaning and one can understand it using simple
cartoons.
Above all, this simple picture makes clear that at least in the language of the t − J model
stripes are formed because quantum-mechanical delocalization energy is gained, analogous
to the kinetic energy gain causing antiferromagnetic superexchange interactions between the
spins in the Mott-insulator. The above consideration has in common with the mean-field
theories discussed in the previous sections that the stripes are caused by severe microscopic
quantum fluctuations. This is in marked contrast with the frustrated phase separation
32
mechanism discussed in section 2.7: this is just based on potential energy. Consider the
row in fig. 11a,b where the hole moves, but now in isolation. One finds that a spin domain
‘wall’ (or kink) is left at the origin while the object which moves is dressed up by the spin
anti-kink. The spin-kink carries S = 1/2 and no charge, while the moving object carries
charge but no spin. These turn out to be quite accurate representations of the spinons
and the holons doing the real work in one dimensional physics. One could now view figure
(11a,b) as an illustration of the local tendency in 2D towards the 1D type spin-charge
separation. However, the frustrations in the spin-system translate into forces ‘confining’ the
holons in strings. The antiphase-boundariness of the stripes is then interpreted as the 2D
manifestation of the topological nature of the 1D excitations.
The question to what extent quantum physics is important for the formation of stripes
is obviously important and it is still not completely settled. One immediately infers from
the simple cartoon picture that the antiphaseboundariness is at the heart of the quantummechanism: the kinetic energy is only released when the stripe is a domain wall. How
to make it accessible for experimentalists? A quantitative measure for the importance of
the quantum processes is the exchange interaction J 0 between the spins located at opposite
sides of the stripe. This interaction is surely mediated by the quantum motions of the holes;
Pryadko et al. (1999) even proved that the antiphaseboundary property cannot be caused by
long wavelength physics. As the spin-wave calculations indicate (Zaanen and van Saarloos
1997, Batista, Ortiz and Balatsky 2001), a maximum in the spin-wave dispersions should
occur at the (π, π) point at a scale set by J 0 and both the data by Lee et al. (2002) on
the nickelate and by Mook et al. (2002) in the underdoped 123 system indicate that this
number is large, of order of the superexchange at half-filling.
The picture by Prelovs̆ek and Zotos (1993) inspired quite a number of other studies. It
forms the starting point of a considerable effort aimed at the understanding of the dynamical
stripes, which is beyond the scope of this review. Let us just mention the lattice string models
constructed by Eskes et al. (1996, 1998) (see also Zaanen et al. 2001, Hasselmann et al. 1999
and Dimashko et al. 1999) to study generic features of the kink-driven stripe delocalization.
33
Another issue is, how to deal with half-filled or even metallic stripes in this language? The
Prelovs̆ek-Zotos picture addressed empty stripes, and it is much less clear how to think about
‘populated’ domain walls. Various interesting ideas have emerged. An intriguing question
was asked by Nayak and Wilczek (1996): accepting that empty stripes are bound states of
holons, why is it so that half-filled stripes are bound states of electrons (i.e. equal amounts
of charge e and S = 1/2 per unit length)? Zaanen, Osman and van Saarloos (1998) assumed
the half-filled stripes in the underdoped regime (x < 1/8) to be insulating because of a 4k F
internal density wave, to subsequently consider the possibility that this state is doped with
additional holes for x > 1/8. They argued that this doped stripe should renormalize into a
three component Luttinger liquid, characterized by additional bosonization fields associated
with the transversal motions of the stripe, next to the usual spin- and charge fields. Bosch,
van Saarloos, and Zaanen (2001) suggested that these transversal kinks associated with the
dopants could be responsible for the ‘Y-shift’ discussed in section 5.
Yet another approach has been followed by Tchernyshyov and Pryadko (2000) and
Chernyshyev, Castro-Neto, and Bishop (2000). The starting point is the t − J z model,
but the focus is initially on low hole density: an Ising spin system is considered with a domain wall present and the dynamics of a single hole is investigated. When the ‘hole’ moves
along the domain wall it turns into a holon, dressed up by a transversal kink. It can only
leave the domain wall by taking up spin to turn into the usual spin-polaron quasiparticle.
Chernyshev et al. conjecture that one can take the single hole propagator of the dilute limit
to calculate what happens at high on-wall hole concentrations. This is further substantiated
by numerical (density matrix renormalization group) calculations by Chernyshev, White,
and Castro-Neto (2002), arriving at the important conclusion that the superconductivity
and quantum spin fluctuations can be viewed as processes which are somewhat secondary
compared to the ‘primary’ t − Jz physics causing the stripes. We also mention the quantum
Monte-Carlo study by Riera (2001), quantifying the scale where the antiphaseboundariness
sets in. Finally, Zachar (2002) challenges the assertion by Chernyshev et al. that domain
walls are stabilized in the t − Jz model at low on-wall hole density. He is arguing instead
34
that at very low hole concentrations in phase domain walls are more stable.
2.6. The ‘exact’ numerical approaches.
The theory of strongly interacting electrons cannot be controlled by theoretical means
and the only way to achieve definite results is often by numerical means. This is not an easy
task either. One can attempt a direct diagonalization on a finite size cluster. The problem
is that the size of the Hilbert space grows very fast (exponentially) with the system size
and the largest system one can handle in this way is a t − J model defined on a 20 site
cluster. Although exact diagonalization studies have been quite important in the context
of the problem of one hole in the antiferromagnet (see Dagotto 1994), the largest exact
diagonalization clusters are too small to address stripes meaningfully. The intrinsic length
scales are of the same order as the linear dimension of the cluster and the physics gets lost
in finite size effects.
A breakthrough occurred some years ago due to White and Scalapino, using the density
matrix renormalization group (DMRG) method invented by White (1992). ‘Renormalization
group’ is a bit of misnomer: this should be seen as a very efficient method to isolate the states
in Hilbert space which are of importance to the ground state. One dimensional problems
can be solved numerically to any desired accuracy. However, two dimensional t − J type
problems are at the frontier of this technology and the method can not be straightforwardly
applied. Nevertheless, White and Scalapino (1998a, 1998b, 1999, 2000a, 2000b), presented
a rather convincing case that the ground state of the t − J model in the physically relevant
parameter regime is about stripes. As we will discuss in some length, these stripes are quite
like the entities one would expect in between the large-S and large-N limits.
With the DMRG method quite large systems can be handled and a typical example is
shown in fig. 12, with dimensions exceeding the stripe length scales (the stripe width, and
separation) sufficiently. White and Scalapino (1998a, 1998b) discovered stripes accidently,
in their exploration of the DMRG model first on t − J ladders and later on the 2D t − J
35
planes. Stripe patterns emerged spontaneously in these calculations. This refers in first
instance to the charge order; to find antiferromagnetism they typically start out with a Néel
order at one of the boundaries of the system which does not disappear when the DMRG is
iterated to completion. This is at the same time indicating the limitation of the method.
They consider a system with a finite size and in a finite volume symmetry breaking is not
possible: the true ground state should be charge uniform and a spin singlet, characterized
by a finite energy gap. However, one can argue that the symmetry restoring fluctuations
involve small numbers for the sizable systems they consider and these small numbers can
get lost in the truncation procedure underlying the DMRG method. In a recent study it is
shown how these problems can be avoided, see White, Affleck and Scalapino (2002).
The results of the DMRG calculations are discussed at great length in a number of papers:
White and Scalapino (1998a, 1998b, 1999, 2000a, 2000b). The (latent) d-wave superconductivity and the stripes are clearly competing, although both can be viewed as ordered
states of the same d-wave pairs, which can be identified in the calculations. This competition can be illustrated using the the DMRG calculations by extending the t − J model with
a next-nearest-neighbor hopping t0 . Pending the sign one either helps the superconductor
or the stripes; this should not be taken to the literal because the physical sign (t 0 /t < 0)
promotes the stripes. We refer in this regard to recent numerical studies by Martins and
coworkers (2000, 2001), exploiting this to study the stripe tendency in more detail, adding
further credibility to the notion that the driving force is the topological ‘holons on a row’
mechanism.
A controversial aspect is the competition between the stripes as found by White and
Scalpino and the phase separation tendency in the t − J model, see also Carlson et al.
(2002). For the two dimensional system, Hellberg and Manouskasis (1997, 1999) have presented a case based on a variational Ansatz that instead of forming stripes the t − J model
would be prone to phase separation in hole-rich and hole-poor matter at all values of t/J.
This gets further support by the proof of Carlson et al. (1998), based on a large-D expansion for the t − J model, that phase separation will occur always in sufficiently large
36
dimensions. There are several caveats. Although DMRG surely has its limitations, the
Hellberg and Manouskasis work is also approximate because it does involve a fixed-node
approximation to avoid the minus sign problem. This is a quite uncontrolled affair, and at a
minimum one should investigate what happens when one starts with an Ansatz incorporating stripe order. In fact, this was accomplished in the paper where the lattice generalization
of the fixed-node Quantum Monte Carlo method was introduced. Van Bemmel et al. (1994)
showed that although the stripe order of the Hubbard model was reduced as compared to
the mean-field solutions it still survived in their quantum Monte Carlo ground state. More
seriously, it seems that the phase separation is interpreted as if the system falls apart in a
half-filled insulator and a featureless Fermi-gas like hole-rich state. Given that the DMRG
total energies and those calculated by Hellberg and Manouskasis are virtually the same,
White and Scalapino (1999a), this seems unlikely, because it would necessarily involve a
highly accidental fine tuning. The length scales associated with the White-Scalapino stripes
are small, implying that sizable energies are involved in the stripe formation. How can these
be nearly identical to those of a state which is microscopically completely different, i.e. the
featureless Fermi-gas? It can be well imagined that the Hellberg-Manouskasis states involve
strong stripe-like microscopic correlations which are hidden from the static properties The
only way to settle these matters is by employing either the multiparticle correlation function as introduced by Prelovs̆ek and Zotos (1993) or the non-local ‘topological’ correlators
introduced by Zaanen and van Saarloos (1997), discussed in more detail in Zaanen et al.
(2001).
Even when stripes are formed, one cannot avoid phase separation in the t − J model because of the attractive Casimir forces mediated by the spin-waves, as discussed by Pryadko,
Kivelson and Hone (1998) and Zaanen et al. (2001). These are however quite weak forces,
which will be easily overwhelmed by the long range Coulomb interactions. The long-range
Coulomb interaction should be included anyhow, and in doing so Arrigoni et al. (2002)
arrived at the important observation that the long range repulsive actually enhance the
superconducting correlations. The reason is likely that the stripes of the bare t − J form
37
at least on the ‘DMRG level’ internal charge density waves, see Bosch, van Saarloos and
Zaanen (2001). The long range interactions will tend to destroy this on-stripe charge commensuration, with the effect that the d-wave pairs trapped by the stripes start to move more
easily.
2.7. Frustrated phase separation.
Up to now we have focussed on quantum theories for the origin of stripes. However, there
is another way of viewing the stripes which seemed to be rooted in a completely different
physics: the idea of frustrated phase separation. This emerged in roughly the same era
as the Hartree-Fock stripes, with main advocates Emery and Kivelson (1987, 1988, 1990,
1993, 1995). Although initially perceived as competing explanations, in the course of time
it became increasingly clear that complementary aspects of the physics are involved. As we
already emphasized in the previous section, the Coulomb interaction has a long range tail
which is just neglected in the standard models. Frustrated phase separation revolves around
the idea that long range interactions can give rise to novel phenomena in highly correlated
electron systems.
The principle is of an elegant simplicity. The prevailing viewpoint is that high-T c superconductivity is about pairing and therefore about rather strong, short range attractive
forces. When the strength of the interactions becomes of order of the bandwidth, the Cooper
instability protecting the pairs looses its influence, and instead the attractive interactions
drive an infinite particle bound state. The system phase separates. When these particles
are electrically charged this in turn leads to a Coulomb catastrophe: the Coulomb energy
would diverge, and the phase separation is ‘frustrated’. Hence, the system has to compromise between the long range repulsions and short range attractions and textures will appear
with a characteristic scale set by this balance.
Initially, Emery and Kivelson (1990) focussed on a fluctuating inhomogeneity in the
form of droplets because this is the generic outcome of frustrated phase separation in the
38
continuum. However, studying a lattice gas model it was discovered by Löw et al. (1994)
that commensuration effects change the physics drastically. This lattice gas model takes the
shape of an Ising spin 1 model on a square lattice where Si = +1 and Si = −1 are interpreted
as hole-rich and hole-poor phases, respectively, while S0 corresponds with charge neutrality.
The Hamiltonian is,
H=K
X
j
Sj2 − L
X
Si Sj + Q
<ij>
X Si Sj
i<j
rij
(5)
K corresponds with the thermodynamics while L, Q > 0 parameterize the short range
attractions and the long range Coulomb repulsion respectively. Löw et al. show that the
ground states of this model are generically stripe-like, consisting of linear domains of hole rich
and hole poor regions. More importantly, this model describes a surprisingly large variety of
phases with nearly equal energies, and accordingly one finds quite a complex phase diagram
as function of the parameters. In combination with the lattice, the competition between
short range attractions and long range repulsions gives rise to dynamical frustration, and
the physics is about an intrinsic tendency to form a glass instead of full order.
With regard to the ‘stripe mechanism’, a consensus has emerged that the quantumaspects and the frustrated phase separation mechanism work in the same direction, involving different aspects of the physics which are likely both needed to explain the physics of
the stripes. However, the frustrated phase separation adds an ingredient superseding this
mechanism question: it suggests a mechanism for intrinsic glassiness. A central result is
due to Nussinov et al. (1999) demonstrating that the O(N ) continuum generalization of the
lattice gas model eq. (5) is characterized by an infinite set of degenerate spiral-like ground
states. This has the spectacular consequence that a jump occurs in the ordering transition temperature at infinitesimal strength of the frustrating interaction (‘avoided critical
behavior’).
This might well be a vital ingredient to the part of stripe physics in cuprates which matters most: it is apparently easy to quantum melt the stripe order, maintaining considerable
microscopic stripe correlations. Two ingredients are needed to make possible the quantum
39
melting of a crystalline state: sufficient kinetic energy should be available (e.g., small mass
density), while the crystal should be soft. It might well be that the frustrating influences
coming from the long range interaction cause the stripes to be a particular soft form of
crystalline matter. Alternatively, one might want to view the anomalous ordering dynamics of the static stripes as a manifestation of this intrinsic tendency towards glassiness, see
Westfahl, Schmalian, and Wolynes (2001).
2.8. The role of the dynamical lattice.
The role of phonons in high-Tc superconductivity has been, and continues to be under
intense investigation. Since the emphasis is here on magnetism, let us just restrict the
discussion to some aspects of the electron-phonon coupling which are of direct relevance to
the stripes.
Bianconi and coworkers (2000) introduced ‘stripes’ of a different kind, discussed in an
electron-phonon language. We refer to the literature for a discussion of these stripes. On a
gross level, this school of thought is focussed on bipolarons being the cause of the pairing.
The idea is that two carriers can gain extra energy by sharing the costs of the creation of a
lattice deformation. However, also here one should worry about the absence of protection of
the two particle channel when polaron binding energies become large; a-priori nothing can
prevent the formation of an infinite particle state, either in the form of phase separation or
in the form of charge order. Although challenged by Alexandrov and Kornilovitch (2002),
Kusmartsev et al. (2000a, 2000b, 2001) addressed the physics of the Pekar-Fröhlich model
(i.e., jellium electrons interacting via long range Coulomb, coupled to longitudinal-optical
phonons) at a finite density. They demonstrate that in the case of a static lattice the
electrons want to clump together in arbitrary numbers. However, the Coulomb repulsions
again interfere and the net result is that string like droplets are formed. Effects of this kind
would surely add up to the other mechanisms, rendering stripes to be further stabilized, see
also Busmann-Holder et al. (2000).
40
The hard questions are related to the possible role of dynamical phonons. A much
simpler issue is about the influence of static phonons/lattice deformations on the electronic
stripes. The problems become of a quantitative nature; in these complicated structures
many different deformations exist which might couple in quite distinct ways to the stripe
charge order. Up to now, studies have appeared on strongly simplified Hubbard-Peierls
type models addressing primarily the coupling of the electrons to the hard, metal-oxygen
breathing type phonons. A recent exception is the work by Kampf, Scalapino and White
(2001) focussed on the influence of the LTT buckling deformations on the stripe ordering.
In going from cuprates via nickelates to manganites, it is clear from experiments that the
magnitude of the lattice deformations dressing up the stripes is increasing drastically. Using
the LDA+U method, Anisimov et al. (1992) reproduce this trend which is explained by an
increasing metal-oxygen covalency associated with the hole state. Zaanen and Littlewood
(1994) presented mean-field calculation based on the Peierls-Hubbard model, specifically
aimed at the nickelate stripes. They arrived at the conclusion that for not to strong electronphonon couplings the stripe driven electronic localization and the polaronic localization
effects mutually enhance each other, adding further stability to the stripes. It was also argued
that strong static lattice dressing of the stripes is detrimental for the superconductivity,
because this will strongly increase the effective mass associated with the electronic matter.
In this regard, the difficulty to observe cuprate stripes in lattice experiments might be seen
as a direct consequence of the simple principle that a strong coupling to the (static) lattice
would render the stripes to be quite classical, as seems to be the case in e.g. the nickelates.
This was taken up by Yi et al. (1998), extending it by a calculation of the mode spectrum
(phonons, and electronic collective modes) around the stripe ground state. This is the
starting point of a highly promising development. As discussed in the experimental sections,
strong phonon-anomalies were (re)discovered in cuprates by McQueeny et al. (1999b), and
Lanzara et al. (2001) linked these to the ‘kink’ anomalies seen in photoemission (see also
section 3.6). As it turns out, very similar phonon anomalies are seen in the nickelate,
Tranquada et al. (2002). McQueeney et al. (2000) argued that similar anomalies appeared in
41
the mode spectrum, calculated from the mean-field stripes. Kaneshita, Ichioka and Machida
(2002) presented recently a detailed analysis, explaining the mechanism by which stripes
induce phonon anomalies of the right kind. They show that these originate in mode couplings
between the purely electronic collective charge modes of the stripes and the phonons. The
charge modes become degenerate with the hard breathing phonons at momenta which are
roughly half way the Brillouin zone, and because of this resonance condition a modest
electron-phonon coupling suffices to cause large anomalies. Several groups are at present
extending this type of calculations to address the spin waves, the optical responses, etcetera,
in stripe systems. Given the complexity of the collective mode spectra it is anticipated that
these efforts will be quite instrumental in guiding the experiments.
42
FIG. 6. An electron stripe according to the early mean-field calculations by Zaanen and Gunnarsson (1989). These authors considered a three band Hubbard model, describing a Cu-O perovskite plane, including the oxygen 2p- and the Cu 3dx2 −y2 states. The arrows indicate the spin
density on the Cu atoms in the insulating domains, while the circles represent the probability to
find the excess holes on the oxygens. For convenience, the (small) hole density on the Cu atoms is
not shown. Notice the anti-phase boundariness: upon crossing the stripe the up spins move from,
say, the A sublattice to the B sublattice.
43
FIG. 7. Explanation of the soliton (‘holon’) found in the mean-field solutions of the doped
Hubbard chain, according to Matveenko and Mukhin (2000). The 1D version of the texture indicated in fig. 6 translates into a kink in the staggered magnetization, giving rise to a mean-field
potential trapping the hole. The electronic structure (right panel) is characterized by upper-(UH)
and lower (LH) Hubbard bands, associated with the electrons in the ‘magnetic domains’. The
‘holon’ is associated with a state lying in the middle of the gap, being occupied by the excess hole.
44
UH
EF
LH
k
//
FIG. 8. According to mean-field theory, one can view the stripes in two dimensions as an
extension of the holons of the one dimensional case, see Zaanen and Gunnarson (1989) and Zaanen
and Oles (1996). In first instance one should consider the motions perpendicular to the stripes and
these form mid-gap states of the same kind as in 1D. However, since their energies are the same
moving along the stripe, the holes are delocalized in this direction and a ‘mid-gap’ band is formed
with a dispersion depending on the momentum k// running along the stripe. In the case of ‘empty’
stripes, this mid gap band is empty and stability is gained because the Fermi energy lies in the
gap between the lower Hubbard band and the mid-gap. When the stripes is ‘half-filled’, the mid
gap band is at quarter-filling and this might drive a on-stripe 4kF (fig. 9) or 2kF 1D-like density
wave instability. According to mean-field, states of this kind are at least locally stable.
45
UH
EF
LH
k
//
FIG. 9. Like fig. 8 except that now the stripe is half-filled (note that a filled stripe carries the
same spins as in the domains and is charge neutral). As Zaanen and Oles (1996) pointed out, the
stripe band is quarter filled and one obtains metastable solutions by invoking on stripe 4k F (figure)
or 2kF (see fig. 12) density waves causing a gap at EF inside the 1D like mid-gap band.
46
FIG. 10. The phase diagram of the t − J − V model suggested by Vojta and Sachdev (1999).
The y axis should be interpreted qualitatively: large y is like the large-N (dashed line) where
the calculations are performed and it is asserted that the spin-Peierls order (inset) changes into
antiferromagnetism when N is reduced (M broken). The bars in the insets indicate two-spin
singlets (‘valence bond pairs’), condensing in columnar spin-Peierls at half-filling. As function of
increasing doping δ the columnar spin-Peierls stays intact in the Mott-insulating domains, which
are now separated by bond centered charge stripes which are build from local d-wave like pairs.
These stripes are compressible and accordingly the state as a whole is a supersolid, showing charge
order (‘C broken’) and superconductivity. For increasing doping the stripes move closer together
(compare the two insets on the right), to disappear at a high doping where the system changes
into a uniform d-wave superconductor.
47
(a)
(b)
~
~
~
~
~
~
~
(c)
(d)
FIG. 11. Cartoon pictures explaining the basic reason why stripes are formed according to the
t − J model. When a hole is injected in a 2D S = 1/2 antiferromagnet (a), it leaves behind a
string of flipped spins when its delocalizes, the ‘magnetic string’(b). These violate the anti-parallel
registry of the spins, thereby frustrating the delocalization of the hole. At finite hole densities, the
holes can coordinate their motions in such a way that they help each other to avoid the frustrating
magnetic strings. Start out with localizing the holes on a domain wall (c). An individual hole
can now move sideways without causing a misoriented spin, and by repeating these hops one finds
kinks propagating along the stripe (d) which will cause the stripe as a whole to move freely over
the plane, unimpeded by the spin system.
48
FIG. 12. The stripes of the t − J model as calculated by White and Scalapino (1999a) using the
numerical DMRG method. These are quite like to what one would expect from the interpolation
from the large N limit to the large-S limit. These ‘inherit’ from large-S the antiferromagnetism
of the magnetic domains while these stripes are also antiphaseboundaries. At the same time,
these stripes are bond centered while they are constructed from d-wave like pairs of holes. This
is especially clear from this figure: it is clearly seen that there is a density wave on the stripe,
which is like the 2kF density wave expected from the mean-field theory for half-filled stripe: two
holes - two electrons - two holes, etcetera. The ‘four hole plaquettes’ just correspond with these
pairs and all what remains to be done is to frustrate this internal density wave in order to obtain
a superconductor.
49
3. The magnetic resonance: theoretical aspects
More than anything else, the magnetic resonance is about the ‘magnetism’ of the
cuprates, but now in the regime where the superconductivity is at its best. There is no
doubt that the ground state of these best superconductors (optimally doped 123 and 2212)
is a spin singlet. Hence, by inelastic neutron scattering one obtains information on the
spectrum of spin triplet excitations. In a conventional, weakly coupled BCS superconductor one expects a spectrum associated with the fermionic Bogoliubov quasiparticles. At
energies lower than the superconducting gap the system has to mantain two particle singlet coherence. The Bogoliubov particle-hole excitations correspond with breaking up such
pairs, liberating the S = 1/2 quantum numbers of the constituent fermions, which may form
triplets showing up in the neutron dynamical form factor. In this case, one expects a incoherent excitation spectrum, starting at the BCS gap 2∆. However, already a long time ago
it was pointed out by Bardasis and Schrieffer (1961), that in the presence of any interaction
triplet bound states will be formed in the gap because of the singularity in the BCS density
of states. These bound states are the particle-particle analogues of the excitons occurring
in the particle-hole channel, and can be viewed as triplet Cooper pairs propagating through
the singlet vacuum. However, this binding is expected to be very weak in a conventional
superconductor and nothing of the kind was ever observed until the arrival of the cuprate
superconductors.
3.1. The facts to be explained.
This is a different affair in the cuprate superconductors. The spectrum of triplet excitations is astonishingly rich and the experimental study of the spectrum by neutron scattering
(augmented by NMR and µSR) has been a fertile research. On the gross scale, one finds two
rather distinct features, see fig. 13. The magnetic excitation spectrum in the 214 system is
dominated by magnetic fluctuations occurring at incommensurate wavevectors (see section
50
5). A rather strong, mostly empirical case emerged that these have to do with one or the
other quantum disordered form of the static stripes discussed in the previous section. We
leave this discussion to the experimental sections 5 and 6. On the other hand, this spectrum
looks quite different in the truly high-Tc superconductors like optimally doped 123 and 2212
(sections 8 and 9). Instead of the incommensurate fluctuations one finds a peak which is
both very sharp in energy and momentum, centered at the antiferromagnetic wave vector
(π/a, π/a) at an energy ∼ 40 meV: the magnetic resonance (fig. 13). In the optimally doped
superconductors, the resonance disappears in the normal state while its weight increases
with the strength of the superconducting order parameter. Upon decreasing doping, the
energy of the peak decreases with doping concentration, suggesting that it would end up at
zero energy at zero doping. On the underdoped side, it persists in the normal state although
it becomes quite broad at higher temperatures. Interestingly, in the presence of a magnetic
field it looses weight even above Tc . This suggests that there is still local superconducting
order in the normal state of the underdoped superconductors, emphasizing once again that
the resonance goes hand in hand with the superconductivity. There is a caveat. For technical reasons, the resonance has only been seen in the bilayer 123 and 2212 system where
it occurs only in the odd c-axis momentum channel. It has been argued that it cannot be
excluded that the resonance is a non-generic special effect associated with the presence of
bilayers. It is not seen in the single-layer 214 and the experimental investigation of other
single-layer cuprates is at present hotly pursued (He et al. 2002).
Knowing only about optimally doped 123 and underdoped 214 one would be tempted
to ascribe the resonance and the incommensurate ‘stripe’ fluctuations to completely different kinds of physics. Quite a confusing and controversial issue is that also in moderately
underdoped 123 spin fluctuations appear at incommensurate wavevectors at energies below
the resonance. At first sight these look quite like the stripe fluctuations of the 214 system. However, these incommensurate fluctuations and the resonance show quite a similar
temperature dependence suggesting that they belong together. Adding to the confusion,
it was shown very recently, Mook, Dai, and Doğan (2002), that stripe order is present in
51
strongly underdoped, barely superconducting 123 (YBa2 Cu3 O6.37 ). The neutron scattering
shows very clear signatures of stripe-like incommensurate spin excitations, which is however
interrupted by a small but discernible resonance structure (fig. 13). We will come back to
this deep problem at the end of this section: it makes one wonder that there is something
not quite right regarding the present understanding of both the stripes and the resonance.
In any other regard an appealing and quite detailed explanation of the resonance has
emerged in terms of a relatively conventional fermiology language. The central assumption
is that the physics is ultimately controlled by S = 1/2 fermionic quasiparticles, spanning up
a big Fermi-surface and communicating with the superconductivity via a particle-particle
potential. One can rest on the well developed theory of metal physics, routine calculations
become possible and accordingly the subject is associated with a vast- and sometimes repetitive literature. In subsection 3.4 an overview of this literature will be found, emphasizing
the differences in the interpretation of the physics behind the ‘metal like’ spin excitations.
In subsection 3.5 we will explain in more detail the interpretation of the resonance within
the RPA framework. As an example we will discuss the recent work by Brinckmann and Lee
(1999, 2001) in some detail. The reason for this choice is twofold: within the ‘metal-physics
paradigm’ it appears that the underlying spin-charge separation framework offers the most
economic interpretation of the various aspects of the physics of the resonance. At the same
time, on the technical level there is a considerable overlap with many other proposals and
Brinckman and Lee, being more or less the last word on the subject, present the most
complete analysis.
This work amounts to a possible answer to the question: ‘given that there is superconductivity, where is the resonance coming from’ ? Subsection 3.6 is devoted to the reciprocal
question: ‘knowing about the resonance, what do we learn about superconductivity’ ? We
will discuss the popular view that magnetic fluctuations are responsible for the pairing,
and pay particular attention to the ingenious sum rule constructions suggesting even deeper
connections between antiferromagnetism and superconductivity.
Before turning into specific explanations of the resonance, we will first spend two sections
52
addressing matters from a most general perspective. In subsections 3.4, 3.5 the focus is on
the question: how to create collective modes out of fermionic matter? Given that these
collective entities exist one can subsequently change perspective, to ask the question: what
can be said in general about the quantum physics associated with these collective entities?
This is the theme of ‘fluctuating-’ or ‘competing orders’, which can be attacked with the
mathematical language of quantum-field theory. The resonance appears within this language
as the signal that the superconductor is close to becoming an antiferromagnet as well. On
this level symmetry is all what matters and in subsection 3.2 we will discuss the bare-bone
version which seems particularly successful for the explanation of the competition between
the stripe antiferromagnets and superconductivity in the 214 system, although Sachdev and
coworkers also came up with some quite surprising ideas related to the resonance. In subsection 3.3 we will focus in on the deep ideas of Zhang and coworkers regarding the possibility
that a larger symmetry (SO(5)) is at work.
3.2. Symmetry (I): spin modes and the coexistence phase.
In order to address the nature of the magnetic excitations in the superconductor one
should first address the more general issue how superconductivity and magnetism relate to
each other on the level of their symmetries. Our focus will be on antiferromagnetism; ferromagnets break time reversal symmetry macroscopically and communicate thereby directly
with the superconducting phase and this requires quite different considerations.
It is often said that a (singlet to triplet) spin gap and superconductivity are one and
the same thing. Although this identification is literal in weak coupling, while a spin gap
might be a good idea if one wants to optimize Tc , it is a rather meaningless statement
in general. Except for the subtleties discussed in the next section, the fundamental fact
is that the superconducting- and antiferromagnetic order parameter fields do not talk to
each other at long wavelength. Only their amplitudes communicate, and pending numbers
all combinations of superconducting- and antiferromagnetic (dis)orders have an equal right
53
to exist. One should consider the Ginzburg-Landau-Wilson order parameter theory, resting entirely on symmetry principles. These considerations are so general that they apply
equally well to the magnetic resonance as to the competition of stripe-antiferromagnetism
and superconductivity.
The first question one should ask is, can a true coexistence state of antiferromagnetism
and superconductivity exist, in the sense that the same electrons are responsible for both
orders? Superconducting order breaks the phase symmetry associated with electrical charge
(U (1)c ) while antiferromagnetism breaks internal spin rotational symmetry (SU (2) s ) and the
overall symmetry is U (1)c × SU (2)s and this is all what one needs to conclude immediately
that a coexistence phase is possible. Although discussed by various authors in the past (e.g.,
Balents, Fisher and Nayak (1998) and Zaanen (1999b)), the subject got a new impetus by
the recent observation of magnetic field induced antiferromagnetism- (Lake et al., 2002,
2001) and stripe-charge (Hoffman et al., 2002a) order, see section 4 and 9. We follow here
Zhang, Demler and Sachdev (2002), and Polkovnikov, Vojta and Sachdev (2002), but see
also Sachdev and Zhang (2002), Kivelson, Lee, Fradkin and Oganesyan (2002), and Chen
et al. (2002). Also notice that the considerations apply equally well to the coexistence of
charge- and superconducting order.
The antiferromagnetic order can be described in terms of a ‘soft-spin’ vector field Φ α and
the superconductivity by the complex scalar field Ψ, with the Euclidean GLW actions, in a
standard notation,
SΦ =
SΨ =
Z
Z
h
dd xdτ (∂µ Φα )2 + m2Φ |Φ|2 + wΦ |Φ|4
h
i
dd xdτ |(∂µ − iAµ )Ψ2 |2 + m2Ψ |Ψ|2 + wΨ |Ψ|4 + Fµν F µν
i
(6)
describing the (independent) dynamics of the superconducting and antiferromagnet orders.
Given the U (1)c × SU (2)s symmetry, the lowest order allowed coupling between spin and
charge is,
Sint = g
Z
dd xdτ |Φ|2 |Ψ|2
54
(7)
Taking both m2Φ and m2Ψ strongly negative it follows immediately that a coexistence phase
is present where both Ψ and Φ are condensed. Since eq. (7) is a pure amplitude coupling,
this state carries a normal antiferromagnetic Goldstone boson (spin wave), as well as a
normal Higg’s boson. Although the amplitudes of both order parameters communicate,
their phase/direction is decoupled.
The circumstance of relevance to the resonance is the one where m2Ψ << 0 while m2ef f =
m2Φ + g|Ψ|2 > 0. After continuation to real time this translates into a spin-triplet mode with
a mass gap, dispersing like ωq ∼
q
c2 q 2 + m2ef f (c is the spin-wave velocity). The correlation
length associated with this spin gap ξM = hc/m2ef f has the following meaning: at distances
shorter than ξM and times shorter than h/m2ef f the system rediscovers that it is actually in
the coexistence phase.
Obviously, this interpretation is quite different from what one gets from BCS theory. In
the weak coupling limit the spin-gap is associated with the pairing gap, meaning that it
corresponds with the energy to break up the singlet Cooper pair, while the characteristic
(coherence) length ξc = hvF /(2∆) is the size of the Cooper pair. Is there an experimental way
to distinguish between both possibilities? This is the punch line of the recent vortex state
experiments in 214. In the neighborhood of the vortices the superconductivity is suppressed
and m2ef f → m2Φ . Assuming that m2Φ is negative, while m2ef f is positive merely because of the
coupling to the superconducting amplitude, patches of antiferromagnetism will surround the
vortices. Since the characteristic length scale ξM is large by default these will percolate in
a long range ordered antiferromagnet already at a small density of vortices. It appears that
along these lines the observations can be explained in a quantitative detail. Alternatively,
when m2Φ is small but positive, the vortex lattice will translate into a potential seen by the
triplet excitations and this causes states to appear in the spin-gap of the system in zero
field. This explains in quantitative detail the inelastic neutron-scattering observations in
the vortex state of optimally doped (x=.15) 214.
The above refers exclusive to the stripe antiferromagnetism of 214 and there is no a-priori
reason to interpret the magnetic resonance of 123 in terms of fluctuating order. Although
55
small as compared to 8J ∼ 1 eV, the magnetic resonance energy (∼ 40 meV) is much larger
than the ∼ 5 meV gap found in 214 indicating that the correlation length is much smaller.
One therefore expects much less, if any, response to magnetic fields of the type found in 214.
However, Sachdev and coworkers discovered a most unexpected way to probe the collective
nature of the physics behind the resonance. It was established by the Keimer group that
already at a low concentration, Zn impurities are remarkably effective in broadening out
the magnetic resonance. Zn impurities might in first instance be considered as missing elementary spins. Sachdev, Buragohain and Vojta (1999) studied the effect of such magnetic
defects when the spin system is close to the quantum phase transition from the quantum
paramagnet (characterized by a magnetic resonance) to the ordered antiferromagnet. They
demonstrated that in 2 space dimensions the physics of the impurities becomes universal.
Among others, they show that the energy width of the triplet mode only depends on the
impurity concentration ni , the spin-wave velocity c and the spin gap ∆ as Γ = ni (h̄c)2 /∆
and their estimate turns out to be remarkably close to the experimental outcome (see section
8). This theory only makes sense when the resonance energy is small as compared to the
ultraviolet cut-off (i.e., the resonance scatters off impurities like a critical mode), in turn
implying that the resonance signals fluctuating order.
3.3. Symmetry (II): SO(5) and the π particles.
The considerations in the previous subsection rest on the assumption that the U (1) c ×
SU (2)s symmetry is in charge. This is quite natural, because it is after all an exact symmetry of the Schrödinger equation. However, as the experience in e.g. nuclear physics
demonstrates, it might happen that symmetry is generated dynamically: the collective behavior of many particles might be more symmetric than that of a few particles. Zhang
(1997) discovered a most natural generalization of this kind in the present context: SO(5),
the smallest Lie group containing Uc (1) × SU (2)s as a subgroup. Only when this symmetry
is exactly realized the macroscopic physics is truly different. Although there is no evidence
56
whatever for this to be the case, a different matter is that the system can get quite close
to it, while it eventually flows back to U (1)c × SU (2)s at long distances due to a relatively
small ‘anisotropy’ energy. Compare it to a Heisenberg spin system subjected to a small
Ising anisotropy. The long wavelength physics will be governed by Ising symmetry, but
upon investing an energy exceeding the Ising anisotropy one will recover the Heisenberg
spin dynamics. In the light of the present evidences such a role of SO(5) cannot be excluded: it could well be that the resonance as a relatively ‘high energy’ excitation has to
do with SO(5). Zhang and coworkers derived a phenomenology starting from mildly broken
SO(5) which is quite similar from the U (1)c × SU (2)s case. In fact, the first mention of antiferromagnetism associated with the superconducting vortex cores was inspired by SO(5),
Arovas et al. (1997), and emerged from earlier ideas by Demler and Zhang (1995) regarding
the nature of the resonance.
As explained in the above, under the rule of U (1)c ×SU (2)s superconducting- and antiferromagnetic order are just independent. The basic idea behind the SO(5) construction, Zhang
(1997), is that they are just two manifestations of the same underlying unity. SO(5) is the
symmetry associated with the rotations of a vector of fixed length in a 5 dimensional space.
Ten different Euler angles are needed and so the SO(5) has ten different generators. The
U (1)c ×SU (2)s group provides 4 generators, and therefore there are six additional generators
available and these correspond with the six ways to ‘rotate’ the antiferromagnet in the superconductor and vice versa: the π modes. Imagine a vector (‘superspin’) having two entries
corresponding with the real and imaginary part of the d-wave superconducting order parameter, while the other three entries corresponds with the three (x, y, z) directions of staggered
spin. The total charge Q is the generator of rotations of the superconducting phase. The
generators of the antiferromagnet ordering at wavevector Q are are Sα =
with α = x, y, z. Consider now the six operators πα† =
P
p
P
†
α
p cp+Q,i σij cp,j
g(p)c†p+Q,i (σ α σ y )ij c†−p,j , with
g(p) = cos(px ) − cos(py ) being the d-wave form factor. It is easily checked that these in total
10 operators (Q, Sα , {πα† , πα }) commute like the generators of the SO(5) group, rotating the
superspin ~n in its 5 dimensional space.
57
Imagine now that the long wavelength dynamics could be described in terms of strongly
renormalized quasi-fermions, such that the Hamiltonian could be written exclusively in terms
of combinations of the 10 SO(5) operators. The dynamical algebra of the problem would
become SO(5) and the spectrum of collective modes would change as compared to the simple
U (1)c × SU (2)s case. For instance, consider the spin singlet superconductor. Under the rule
of U (1)c × SU (2)s one would find a single massless phase mode and a triplet of massive
spin excitations. In SO(5) one would find besides the phase Goldstone mode a total of 9
massive excitations corresponding with linear combinations of spin- and π excitations. If
one of these modes would soften, a phase transition would follow to a state where a π mode
is condensed. Since π modes rotate the superconductor into the antiferromagnet (and vice
versa) this state corresponds with a superconducting-antiferromagnetic coexistence phase.
Finally, when all SO(5) operators would appear on the same footing in the Hamiltonian so
that the system becomes invariant under SO(5) rotations, the higher symmetry would be
fully realized. Zhang (1997) speculated that the cuprates could approach this symmetric
point sufficiently closely that one can profit from the inherent simplifications associated with
the larger symmetry.
The problem has been all along in the microscopic definition of the SO(5) operators.
In the original proposal, the fermions were taken to be literal electron operators, while the
antiferromagnet was associated with the Mott-insulator. It was immediately recognized
that this does not make sense, Greiter (1997) and Baskaran and Anderson (1998): how to
connect smoothly a Mott-insulator with an energy gap of 2 eV with a superconductor being
charge-wise infinitely soft? However, soon thereafter numerical evidences appeared in favor
of a SO(5) algebra at work in the t − J model. By studying the spectral functions of the π
operators, Meixner, Hanke, Demler and Zhang (1997), and Eder, Hanke and Zhang (1998)
demonstrated that the excitation spectra of small clusters could be categorized in terms of
approximate SO(5) multiplets.
This puzzle was at least in part resolved by work on t − J ladders, Scalapino, Zhang
and Hanke (1998). Instead of constructing the operators in momentum space one should
58
use a strong coupling, real space representation. The spin operators are associated with
the electrons of half-filling turning into on-site spins due to the Mott-constraint: S i,α =
P
†
α
γδ ci,γ σγδ ci,δ .
The Cooper pairs, on the other hand, cannot exist on a single site because of
the strong repulsions and instead one can consider ‘link’ or nearest-neighbor pair operators:
† †
Lαβ
<ij> = ciα cjβ , having a finite overlap. With the operators used in the 2D t − J model
calculations, Scalapino et al. demonstrated that by choosing a Hamiltonian such that these
link pairs are formed on the rungs of the ladder it becomes easy to construct a manifestly
SO(5) symmetric Hamiltonian. As it turns out, to construct such a Hamiltonian one has
to allow for single site double occupancy (associated with the pair fluctuations) thereby
violating the single occupancy constraint. As Zhang et al. (1999) showed, this problem
could be overcome by a Gutzwiller projection afterwards, leaving SO(5) in essence intact
(‘projected’ or p − SO(5)).
Matters are still not quite resolved in the two dimensional case. Interpreted in terms
of bare electron operators, one finds that the set of link-pair and spin operators do not
close into a SO(5) algebra, as related to the fact that operators associated with different
links can share a site. This problem can be overcome by invoking long range interactions as
shown by Rabello et al. (1998), Henley (1998) and Burgess et al. (1998), but this leads into
Hamiltonians which have not much to do with electrons in solids. Van Duin and Zaanen
(2000) discovered a way to avoid the above difficulties by tailoring a special two dimensional
lattice structure. A similar construction involves plaquette-, instead of link degrees freedom,
Dorneich et al. (2002). The idea is to start out with a square lattice, to subsequently
pull apart the sites in ‘site-plaquettes’. The link-pairs live on the links of the original
lattice while the links of the site-plaquettes only mediate ferromagnetic exchange interactions
between the spins of the electrons residing on the same small plaquette. The link operators
can be combined in a (projected) SO(5) and subsequently a class of Hamiltonians can be
constructed illuminating various aspects of the physics. One arrives at a quite versatile
model, describing a large variety of possible phases, including the antiferromagnet Mott
insulators, singlet d-wave superconductors, but also antiferromagnetism- d-wave coexistence,
59
s-wave and triplet superconductors, etcetera, and even a tricritical point where p − SO(5) is
exactly realized. Quite significantly, one finds that matters reduce to U (1) c × SU (2)s , unless
one incorporates pair hopping processes which frustrate the antiferromagnet. This might well
be a quite general feature: at the moment the motion of the Cooper pairs start to frustrate
the antiferromagnetic correlations in the spin system one cannot avoid an admixture of π
modes in the excitation spectrum.
Let us now return to the discussion of the magnetic resonance. SO(5) started out with
the assertion of Demler and Zhang (1995) (see also Demler, Kohno and Zhang (1998)) that
the magnetic resonance was the experimental manifestation of the π rotation, with its energy
measuring ‘the distance’ between the superconducting ground state and the (commensurate)
antiferromagnet. This suggestion was dismissed by Greiter (1997) and Tchernyshyov, Norman and Chubukov (2001) emphasizing that the resonance has to be a massive spin wave
associated. On closer inspection one finds, however, that these arguments are based on
quantitative considerations regarding particular models having no fundamental status. In
fact, the debate makes not much sense. The issue is that π modes and spin waves are not
mutually exclusive: it is by principle a matter of degree. In a singlet superconductor there
are two ways of exciting a massive spin triplet at (π, π): either by spin flips or by a π operator. Except for the SO(5) symmetric point, the spin waves and the π modes are symmetry
wise indistinguishable, meaning that these modes are always coupled, see for instance van
Duin and Zaanen (2000). The strength of this mode coupling is governed by microscopic
dynamics and it is therefore impossible to make decisive theoretical statements. The only
way to decide the issue is by experiment. The information on the spin content, as derived
from neutron scattering, does not suffice. One needs to know in addition the spectral function associated with the π operators Sπ , and this information is at present not available, see
Bazaliy, Demler and Zhang (1997).
3.4. Restoring the fermions: the various viewpoints.
60
The focus in the above two sections has been entirely on the collective fields. However,
eventually these collective fields are made out of electrons and it is a-priori unclear if the
energy scale where the electrons are ‘eaten’ by the order parameter fields can be taken to be
infinitely large. To the contrary, photoemission and tunneling measurements demonstrate
that degrees of freedom having a finite overlap with the bare electron exists all the way
down to zero energy: the nodal fermions. Since the electron carries S = 1/2 this implies
the existence of low energy spin degrees of freedom ‘derived’ from the nodal states. In
addition, in the overdoped regime a reasonable case can be made that a conventional BCS
type behavior is re-established, such that the onset of superconductivity goes hand in hand
with the formation of the electron pairs. Last but not least, just observing the empirical
facts pertaining to the resonance itself, it is clear that one misses important pieces of the
physics if one takes only into account the workings of the collective fields. Why is the
resonance disappearing above Tc , at high dopings? How to explain the incommensurate
‘lobes’ of the resonance seen at energies below the resonance moving away from (π, π) with
decreasing energy (fig. 13)? This dispersion is rather the opposite of what one would expect
for a ‘normal’ massive spin wave.
These facts do find a qualitative explanation in a rather conventional fermiology language. The fundamental assumption is that the vacuum is controlled by a Fermi-surface, at
least associated with the S = 1/2 spin excitations, undergoing a weak coupling instability
into a d-wave superconductor. This is the working horse, producing a successful phenomenological interpretation, leaving still room room for various further interpretations. The most
conservative version is to take it very literally, meaning that the cuprates are interpreted as
Fermi-liquids all along (including the normal state) undergoing a weak coupling BCS instability, while the excitations can be calculated from the leading order in perturbation theory
controlled by the weakness of the residual interactions (random phase approximation, RPA),
see Maki and Won (1994), Lavagna and Stemmann (1994), Mazin and Yakovenko (1995),
Blumberg, Stoikovic and Klein (1995), Bulut and Scalapino (1996), Salkola and Schrieffer
(1998), Abrikosov (1998), and Norman (2001). Given the anomalies of the normal state,
61
and especially the pseudo-gap behavior in the underdoped regime (including the persistence
of the resonance) this is generally perceived as an oversimplification.
At a next step, one can push the perturbation theory to a higher order and the most
successful work of this kind is based on the fluctuation exchange (FLEX) approximation,
where the magnetic excitations are calculated from a single electron propagator, dressed
with a self-energy which is self-consistently calculated from the scattering against the collective fluctuations (Pao and Bickers (1995); Dahm, Manske and Terwordt (1998); Takimoto
and Moriya (1998); Manske, Eremin, and Benneman (2001)). Although not much changes
qualitatively at low temperatures, FLEX gives a better account of the physics at high temperatures. It describes damping of the resonance, while at least a sense of pseudo-gapping
is recovered in the underdoped regime. Although more phenomenological, the work by Littlewood et al. (1993), based on the marginal Fermi-liquid (MFL) phenomenology by Varma
et al. (1989), has a similar attitude. The RPA bubbles are calculated using single particle
propagators dressed up with marginal Fermi-liquid self-energies. Calculations of this kind
have been presented, aimed at an explanation of the anomalous frequency and -temperature
dependences of the incommensurate spin fluctuations in the normal state of 214. Although
this has not been pursued for the superconducting state, one would expect quite similar
answers as for the ‘bare’ RPA given that the anomalous self-energy is assumed to disappear
below the superconducting gap. As an alternative, the work by Onufrieva and Rossat-Mignot
(1995), based on a Hubbard operator technique should be mentioned.
It is widely believed that on a microscopic scale the electrons in the cuprate planes have
to do with a single component electron system subjected to strong on-site interactions (i.e.
Hubbard model and extensions). Given that the system starts out to be very strongly interacting, it is a-priori unclear if perturbation theory in terms of the bare interactions can be
pushed far enough to produce meaningful answers. Instead, one can assert that this difficult
microscopic dynamics renormalizes into some effective theory at intermediate scales which
can subsequently be used to address the low energy phenomena. This underlies the idea of
Pines and coworkers of the ‘nearly antiferromagnetic Fermi-liquid’, see for instance Mon62
thoux, Balatsky and Pines (1991). It is asserted that besides the fermion system a sector of
collective, overdamped spin excitations is generated which interacts relatively weakly with
the bath of fermions. Upon lowering temperature, this spin system tends to antiferromagnetic order. In this framework, the resonance is interpreted as the antiferromagnetic mode
becoming underdamped because of the opening of the superconducting gap in the fermion
spectrum, while its decreasing energy as function of decreasing doping is interpreted with
the softening associated to the approach to the antiferromagnetic instability, Barzykin and
Pines (1995), Morr and Pines (1998, 2000a, 2000b). Taking this spin-fermion model as a
starting point, Chubukov (1997) has made the case that one can control the perturbation
theory with unconventional means, making it possible to evaluate the theory even in the
strongly coupled regime. The idea is that under the condition of overdamping of the spin
fluctuations the inverse of the number of hot spots (i.e., the number of crossings (8) where
the magnetic Umklapp surface intersects the Fermi surface) can be used as small parameter
to control the theory. The end result of this strong coupling theory, Abanov and Chubukov
(1999) looks much like the phenomenological modelling based on lowest order perturbation
theory.
On a side, multicomponent models like the spin-fermion model, but also the bosonfermion model asserting that fermions coexist with preformed Cooper pairs (see Domanski
and Ranninger (2001) and ref.’s therein), have to be considered as Ansatzes, ill founded
in the microscopy of the electrons in the cuprate planes. However, quite recently Altman
and Auerbach (2002) reported on a novel type of real space renormalization group (RNG)
procedure which they applied to the t − J model. Remarkably, according to this RNG a
separation between a fermionic sector and a bosonic sector is dynamically generated. The
bosonic sector takes care of the ‘fluctuating order’ aspects both involving the superconductivity and the antiferromagnetism along the lines discussed in sections 3.2 and 3.3, which is
in turn relatively weakly coupled to the fermions.
Finally, there is a class of theories having in common a more radical departure of the
physics of conventional metals and superconductors. These can be grouped in two sub63
families: the ‘hidden order’ theories, and the spin-charge separation gauge theories. The
hidden order theories find their inspiration in the interpretation that the anomalies of the
cuprates are in first instance governed by a form of order present in the underdoped regime,
of a kind which is hard to detect by existing experiments. This order is then assumed to
disappear at a zero-temperature quantum phase transition, governing the physics of the optimally doped superconductors. A typical example of a possible order is based on the idea that
some regular pattern of spontaneous orbital currents could be present: the staggered flux
phase or d-density wave, breaking time-reversal and translational symmetry, Chakravarty
et al. (2001), and the ‘charge transfer’ flux phase by Varma (1999) which is just breaking
time-reversal symmetry. These phases have their own form of ‘magnetism’: the spontaneous orbital currents give rise to a pattern of small magnetic moments which should be
in principle observable. In Varma’s version these orbital moments compensate each other
within the unit cell and he suggested that this should lead to observable consequences in the
polarization dependence of the angular resolved photoemission. His predictions seem to be
confirmed in a recent experiment of this kind, Kaminski et al. (2002). The staggered flux
phase is more easy to detect because it also breaks translational symmetry, and the orbital
moments should be observable in conventional magnetic probes. Mook, Dai and Dogan
(2001) identified a weak antiferromagnetism which appeared to be of the right kind. However, follow up experiments by Sidis et al. (2001) showed that the associated moments lie in
the planar directions, consistent with ‘conventional’ spin magnetism and hard to explain in
terms of orbital currents, expected to cause moments pointing in a perpendicular direction.
As was argued by Tewari, Kee, Nayak and Chakravarty (2001), the persistence of the magnetic resonance in the underdoped normal state is easily explained in this framework: the
d-density wave order persists above Tc and this causes a gap in the quasiparticle spectrum
which is sufficiently similar to the superconducting gap to cause a resonance along the lines
explained in the next section.
Besides the flux phases, other suggestions for hidden order invariably involve the even
more exotic physics associated with gauge theories. A most recent idea by Zaanen et al.
64
(2001) further elaborated by Zhang, Demler and Sachdev (2002), and Polkovnikov, Vojta
and Sachdev (2002) is based on the notion that the domain-wall property of the stripes can
persist into the otherwise featureless, spin- and charge disordered superconductor. At some
doping also this ‘topological order’ should be destroyed, and using symmetry arguments
it can be demonstrated that this ‘stripe fractionalization’ transition is governed by the
confinement transition of Ising gauge theory. This theory is not yet understood to an extent
that meaningful statements regarding the resonance can be made.
This brings us finally to the gauge theories of spin-charge separation being the subject
of the next subsection.
3.5. The resonance peak and spin-charge separation.
In the spin-fluctuation theories as discussed in the previous section, the basic outlook
is that the zero-temperature ground state is much like a conventional BCS superconductor.
However, the underlying Fermi-liquid is strongly interacting in the sense that interaction
effects switch on much more rapidly when one goes up in energy or temperature, and the
magnetic resonance is a signature of the importance of these interactions. Upon raising
temperature, the same interactions start to take over and their main effect is that they
‘mess up’ the Fermi-liquid, causing propagators to become overdamped. The normal state
is perceived as a rather structureless affair.
Given its remarkable regularities, one could wonder if there is not more behind the physics
of the normal state. Given that the superconducting state is derived from this normal state,
a hidden structure in the latter could also be quite consequential for the nature of the
mechanism which causes the anomalously large stability of the superconductor. In the
hidden order theories discussed at the end of the previous section it is just assumed that
some other order is around which is just hard to see. The gauge theories of spin-charge
separation are conceptually more subtle: there is only a single superconductor, but the
competitors of the superconducting state are exotic and they leave their mark in the normal
65
state. Mathematically, these matters are controlled by gauge principle and these states are
then believed to carry elementary excitations corresponding with pieces of the electrons.
The idea was introduced early on by Baskaran, Zou and Anderson (1987), and it evolved in
the course of time to a high level of sophistication (e.g., the Z2 version of Senthil and Fisher
(2000), and the SU (2) theory by Wen and Lee (1996)).
The magnetic resonance was already addressed in this language quite some time ago
by Tanamoto, Kuboki and Fukuyama (1991), followed by a number of other studies: Zha,
Levin and Si (1993), Tanamoto, Kohno and Fukuyama (1993, 1994), Stemmann, Pépin, and
Lavagna (1994), Lercher and Wheatly (1994), Liu, Zha and Levin (1995), and Normand,
Kohno and Fukuyama (1995), culminating in the most complete recent work by Brinckmann
and Lee (1999, 2001). The last work is especially quite insightful regarding the kinematics
behind the RPA which is also of relevance to the theories discussed in the previous section.
In the remainder of this section we will discuss this the Brinckmann-Lee version in more
detail.
We introduced the idea of spin-charge separation already in section 2.3. In fig. 14 the well
known phase diagram is reproduced. The basic idea is that the electrons fall, at least approximately, apart in bosons carrying the charge of the electrons and fermions representing
the spins of the electrons. The former (‘holons’) just behave like hard-core bosons occurring
at a density proportional to the chemical doping x, and they Bose condense at a temperature
TBE ∼ x. The fermions, on the other hand, span up a large Fermi surface which is thought
to represent the spin system and its ‘spinon’ excitations. These are subjected to a d-wave
BCS like interaction which causes an instability at a temperature Td into a ‘spinon superconductor’. As in a normal superconductor, the ground state is a spin-singlet and the nodal
states now correspond with pure S = 1/2 excitations. This spinon condensation temperature
would be at maximum at half-filling, if the competing antiferromagnetic order would not
interfere, and it decreases with increasing doping because of the decreasing volume enclosed
by the spinon fermi-surface. The superconductor corresponds with the state where both
spinons and holons are condensed: such a state is symmetry wise indistinguishable from a
66
BCS superconductor. This produces clearly a maximum of Tc at an ‘optimal’ doping, see
fig. 14. However, in the underdoped regime the spinons condense at a temperature which is
higher than Tc and this corresponds with the opening of a d-wave like spin gap. This is the
simple interpretation of the pseudogap phase in this framework. From a phenomenological
point of view it is quite appealing. Because the pseudo-gap has a great effect on the magnetic properties and the single electron properties, while its effects on charge transport are
quite moderate, one could directly conclude that a spin-charge separation should be at work
of this kind.
The starting point is the t − J model and the slave formalism which we already discussed
in the large-N context in section 2.4. There is no doubt that the correct saddlepoints
in the large-N limit are the ones discussed by Vojta et al. (1999, 2000), with the main
difference that the spin system becomes a spin-Peierls state instead of the massless d-wave
state. However, one could argue about the relevance of these results when N = 2. The
only way to proceed is by treating exactly the gauge fluctuations, needed to describe the
locality of the constraint and this is a very difficult task. Instead one can in first instance
take a phenomenological position and assert that a saddle point which is metastable at large
N might become relevant for N = 2 for no other reason than that it looks more like the
experiments. Given the present state of the gauge theory, the status of the phase diagram
(fig. 14) is no more than an educated guess.
Omitting these gauge fluctuation-difficulties, the bottom line is that according to the
‘classic’ spin-charge separation scenario the magnetic properties can be derived from a system
of weakly interacting spin-full fermions, characterized by a big Fermi-surface with a volume
∼ 1 − x and a tendency to form a d-wave ‘superconductor’. This is the same basic problem
underlying most of the theories discussed in subsection 3.4. The difference is that the
‘spin-physics’ of the d-wave superconductor sets in at the pseudogap temperature which
is unrelated to the superconducting transition temperature. In the magnetic fluctuation
scenario’s discussed in section 3.4, Tc and the pseudo-gap temperature are one and the same
thing on the saddle-point level, and they have to be ‘disconnected’ by tedious perturbation
67
theory.
Let us now turn to the details of the RPA analysis. RPA (or ‘bubble summation’, ‘time
dependent Hartree-Fock’) has the same status as the use of classical equations of motion to
derive spin waves, but now applicable to fermion systems. The well known result for the
dynamical magnetic susceptibility is,
χ(~q, ω) =
χ0 (~q, ω)
1 + J(~q)χ0 (~q, ω)
(8)
where J(~q) is the effective interaction (becoming here 2J[cos(qx ) + cos(qy )], where J is the
superexchange constant) while χ0 is the Lindhard function associated with the fermions, see
also eq. (19). In the presence of a BCS pair potential, this becomes
χ0 (~q, ω) =
ε~ ε~ + ∆~k ∆~k+~q
1
1 + ss0 k k+~q
8
E~k E~k+~q
s,s0 =±1
X X
~k
×
"
f (s0 E~k+~q) − f (sE~k )
ω + sE~k − s0 E~k+~q + i0+
where f is the Fermi-function, ∆~k the gap function, E~k =
#
(9)
q
ε~2k + ∆~2k the dispersion of
the Bogoliubov quasiparticles and ε~k the band dispersion. The imaginary part χ000 of χ0
represents the density of states of particle-hole excitations at energy ω and momentum q,
weighted in a BCS superconductor by coherence factors. The real part χ00 can be obtained
from the imaginary part by Kramers-Kronig transformation.
In the normal metallic state, χ0 is a rather uninteresting function both of momentum
and energy. In order to cause it to be interesting one has to be close to a nesting condition
(or van Hove singularity), causing strong enhancements of χ000 at particular momenta ~q0 .
This causes in turn χ00 (~q0 , 0) to become large and negative. χ00 appears in the denominator
of eq. (8), and it might become large enough such that J(~q0 )χ0 (~q0 , 0) → −1. A singularity
appears signalling the instability towards order at wave vector ~q0 . In the approach to the
instability, the real part of the denominator is decreasing, and the peak in χ 000 will be further
enhanced in the full susceptibility χ00 . This is the conventional view on spin fluctuations in
Fermi-liquid metals.
68
In the context of the normal state of the cuprates, it was pointed out by several groups
that it is not unreasonable to expect that the Fermi-surface of the 214 family shows sufficient pseudo-nesting to explain the incommensurate spin fluctuations, see Littlewood et al.
(1993), Si et al. (1993), and Zha et al. (1993). In the mean-time this is taken less serious
√
because a Fermi-surface derived incommensurability should behave like ε ∼ x as function
of doping (because the Fermi surface volume ∼ 1 − x) while the experiments show ε ∼ x
for x < 1/8, to become doping independent at larger dopings, more consistent with a stripe
interpretation. It was also argued that because of differences in the Fermi-surface shape a
broad commensurate response should be expected in 123. In the mean-time this interpretation is superseded by the knowledge regarding the resonance. All together, signs of these
‘Fermi-surface spin fluctuations’ are completely lacking in the normal state at temperatures
above the pseudo-gap temperature.
The situation changes drastically when the superconducting gap opens up. By considering carefully the kinematics, one will generically find that in the presence of the superconducting gap, χ000 (~q, ω) will disappear discontinuously at some ω0 (~q). This should be obvious
for a system of normal fermions with a ~k dependent gap. As an added difficulty one has
to account for the coherence factors in the superconducting state and as it turns out these
are just of the right kind in a d-wave superconductor, while they in fact smooth out the
singularities of the s-wave superconductor. Such a discontinuity in χ000 gives rise to a singularity in χ00 . This in turn implies that 1 + J(~q)χ0 (~q, ω) has two zero crossings, one of which
occurs at an energy ωres (~q) < ω0 (~q). Hence, χ develops a pole inside the gap at ωres (~q),
corresponding with a bound state (sharp peak) inside the gap. This is the basic explanation
for the magnetic resonance in this RPA language. The dispersion of the resonance ωres (~q)
and its ~q dependent spectral weight now become a matter of detailed consideration of the
kinematics, depending in turn on the details of the Fermi-surface and the gap function.
Using the spinons one finds precisely the right Fermi-surface evolution as function of
doping to explain the doping dependence of the magnetic resonance. Let us first consider
the ‘magnetic’ response at ~q = (π, π). At half-filling, the spinon Fermi surface exactly
69
coincides with the (π, π) Umklapp surface, and accordingly the nodal points lie on the
Umklapp surface. It follows immediately that χ000 has finite weight all the way to zero
energy and a bound resonance cannot be formed (instead the system is unstable towards
antiferromagnetism). Upon doping the volume of the Fermi-surface shrinks ∼ 1 − x with the
effect that the nodal points move away from the Umklapp surface and χ000 is zero until one
reaches an energy ω0 which is roughly twice the energy of the Bogoliubov particles at the
hot spots, where χ000 switches on discontinuously, see fig. 15. Upon calculating the full χ one
finds the resonance with an energy ωres (π, π) (fig. 16), which grows roughly proportional to
x and a spectral weight which is steadily decreasing with x. Finally, at large dopings the
peak start to shift downwards: here ωres follows the gap maximum, and according to the
slave theory this maximum shifts down in the overdoped region.
The ‘inverted’ dispersion, i.e. peaks moving away from (π, π) upon lowering energy
(fig. 13), is quite natural in the RPA framework. The results obtained by Brinckmann
and Lee (2001) are shown in fig. 16. At energies above the resonance the influence of
the superconducting gap rapidly diminishes and the χ000 starts to look like the featureless
susceptibility of the normal state. At energies below the resonance the superconducting gap
is of course in charge, but the wave numbers associated with the singularities shift away
from (π, π). Recall fig. 15; the ω = 0 nodal points are connected by a ‘2kF ’ vector which is
smaller than (π, π), and at very low energy one would expect very weak features in χ 00 at
momenta (π ± δn , π ± δn ) with δn being the difference between the nodal spanning vectors
and (π, π): one expects a preferential orientation of the ‘incommensurate peaks’ along the
zone diagonals, inconsistent with the experiments. This ‘flip’ from the diagonal to the
horizontal/vertical direction is explained by Brinckmann and Lee invoking a specialty of the
quasifermion dispersions. In the direction perpendicular to the Fermi-surface, the dispersion
is more steep for momenta k > kF than for k < kF turning the contours in momentum
space of the quasifermion dispersions at a given energy from the generic ellipsoids into
‘banana’s’ (fig. 15) having a flat face oriented parallel to the magnetic Umklapp surfaces.
This corresponds with a dynamical nesting condition giving rise to incommensurate peaks
70
with a horizontal/vertical orientation.
This completes the discussion of the principles behind the RPA theories of the magnetic
resonance: upon ignoring the spinon aspect, the Brinckmann and Lee work is representative
for this school of thought. In detail the different versions do look different, and the outcomes
do depend quite a bit on the details of the bandstructure, effective interactions and so on.
In order to arrive at a convincing comparison with experiment fine-tuning is involved and
one should always be aware of the possibility that RPA could be no more than a form of
curve fitting. We already alluded to the fine tuning in the Brinckmann-Lee calculations
required to get the incommensurate peaks at the right momenta. Even worse, in their case
it turns out that the next-nearest-neighbor hopping t0 has to be fine tuned to change the
nodal ‘ellipsoids’ in the ‘banana’s’ (fig. 15), in turn needed to find a magnetic resonance all
together
Let us finally discuss the striking dependence of the magnetic resonance on the momentum in a direction perpendicular to the planes. According to the experiments, the resonance
appears exclusively in the odd (qz = π/c) channel, while in the even (qz = 0) channel only a
incoherent background is found starting at a large energy (∼ 90 meV). Initially, it was suspected that the presence of bilayers could be essential for the appearance of the resonance,
but given the recent demonstration by He et al. (2002) for the presence of a resonance in a
single-layer thallium compound this seems no longer to be an issue.
Considering the overall energy scales, the situation is not that different from what is
found in the half-filled antiferromagnetic insulator YBa2 Cu3 O6 , characterized by a massless
acoustic magnon in the odd channel, while the spectrum in the even channel is characterized by an optical magnon associated with a counter-precession of the spins in the two layers
having an energy ∼ 100 meV, set by the interlayer exchange. Given the shear size of this
energy, this is about a short distance physics and it can be imagined that this even-odd
difference of the spin system of the insulator can persist up to rather high dopings. Millis
and Monien (1996) discuss the interlayer spin physics in this spirit. In the RPA view, the
physical picture is quite different. Intrinsically the physics at high energies is that of free
71
fermions. To explain the exclusive occurrence of the resonance in the even channel, initially suggestions appeared referring to specialties of the interlayer pairing, see Mazin and
Yakovenko (1995), and Yin, Chakravarty, and Anderson (1997). However, this is not at all
necessary. We again refer to Brinckmann and Lee (2001): for simple reasons the interaction
J(~q) is substantially weaker in the even channel as compared to the odd channel with the
effect that the even channel susceptibility looks much more like the Lindhard function. The
resonance is missing and instead a hump is found in the even susceptibility around 100 meV.
This is just caused by scattering involving the singularities at the gap maximum, which is
reflected in the bare susceptibility.
3.6. The resonance and superconductivity.
As we already discussed, an intriguing aspect of the resonance is its close relationship with
superconductivity: its weight grows with the strengthening of the superconductivity, and it
is suppressed by magnetic fields. The RPA considerations suggest a simple interpretation.
The superconducting gap is needed for the singularities in the bare susceptibility, as required
for the formation of a bound state. In this view, the resonance is just a passive side effect
of the superconductivity. Could there be more to it?
Much work on the resonance is motivated by the spin-fluctuation exchange mechanism
for superconductivity. The idea is that instead of phonons, the spin fluctuations can be taken
as the bosonic glue mediating the attractive interactions between the electrons. This old idea
acquired quite some popularity when it became clear that these are d-wave superconductors:
using general arguments based on the strong momentum dependence of the interaction vertex
between spin fluctuations and electrons, d-wave symmetry is natural, see Bickers, Scalapino
and White (1989), and Monthoux and Pines (1992). In this mindset, advocated by Morr
and Pines (1998, 2000a) and Abanov and Chubukov (1999) the magnetic resonance acquires
a special significance: it is the long-sought for pairing glue becoming directly visible in
experiments. This should not be taken too literally. The energy of the magnetic resonance
72
is less than the superconducting gap maximum and it would therefore act by itself as a
pair breaking mode. Instead, it is envisioned that the magnetic modes responsible for the
pairing are strongly overdamped in the normal state and thereby hard to detect. However,
when the superconductivity sets in a back reaction follows on the spin fluctuations causing
a propagating piece to emerge at the low end of their spectrum – the magnetic resonance.
Given that the magnetic resonance exists and assuming that it has to do with the spinfermion model, it should be that the quasiparticles of the superconductor are scattered by
the resonance. Since the resonance is sharp in energy and momentum, while it is showing
strong temperature and doping dependences, it should be possible to find back the effects
of the scattering in the quasiparticle spectra as measured by photoemission and tunneling.
Subsequently, from these residual effects one could attempt to deduce the properties of the
bare spin-fermion model. Although still controversial, strong claims along these lines are
found in the literature. The ARPES spectra of the superconductors are characterized by a
peak-dip-hump structure at the anti-nodes, Dessau et al. (1992), which has been explained
in terms of a strong scattering to the resonance ∼ 40 meV, Norman and Ding (1998) and
Abanov and Chubukov (1999). Much attention was drawn to the observation of a kink in the
quasiparticle dispersion along the (π, π) direction, Bogdanov et al. (2000). The quasiparticle
velocity increases suddenly upon passing an energy ∼ 50 meV, and this is reminiscent of well
known mass renormalization effects by optical phonons in conventional metals: at energies
less than the mode energy the quasiparticle has to carry around the lattice polarization,
decreasing its velocity, while at higher energies the phonon cannot follow the quasiparticle.
Eschrig and Norman (2000) claim that this kink follows closely the resonance both with
regard to it doping- and its temperature dependence. However, this interpretation was
challenged by Lanzara et al. (2001), especially so by their demonstration that the same
kink occurs in the 214 system where the magnetic resonance is absent. Lanzara et al.
argued that it has to do instead with the anomalous Cu-O breathing phonon which was
discussed in section 2.8.
Recently, an interesting argument started regarding the numbers in the game. Kee,
73
Kivelson and Aeppli (2002) observe that in absolute units the resonance carries quite a
small spectral weight. The total weight I0 =
R
S(~q, ω)d2 qdω/(8π 3 ) turns out to be only a
few percent of the local moment sum rule h̄2 S(S + 1) = h̄2 (3/4). Assuming a spin-fermion
~ · ψ †~σ ψ, the leading contribution to the electron self-energy is of
coupling of the form g S
order I0 g 2 /ωres , less by a factor I0 as compared to a naive estimate. This is just a pole
strength effect: during 1 − I0 of its time the magnetic resonance is not behaving like the
bare magnetic mode. As a consequence, in order to have a substantial influence on the
quasiparticles the coupling g has to very large. This coupling should be estimated using
independent information, and Kee et al. arrive at g ' 10 meV, while Abanov et al. (2002)
find g = 1 eV starting from different assumptions!
Much of the above is based on model assumptions which cannot be claimed to be universally valid. A different matter is the recent discovery of a number of sum rules associated
with the properties of the superconductors, having a much more fundamental status. These
connect properties of the superconducting state to changes occurring in the dynamical magnetic susceptibility, with the magnetic resonance playing a central role. A first sum rule
is due to Scalapino and White (1998). They observed that if one can split off a magnetic
interaction J
P
<ij>
~i · S
~j in the effective Hamiltonian describing the physics at intermediate
S
length scales (e.g., the t − J model), one can relate the gain in magnetic energy ∆E J associated with the superconducting order to the difference between the dynamical form factors
of the zero temperature normal state (SN , e.g. above Hc2 ) and the superconducting state
(SS ) by,
3 Z π/a 2 Z ∞ d(h̄ω)
dq
[(cos(qx ) + cos(qy ))(SN (~q, ω) − SS (~q, ω))]
∆EJ = J
2 −π/a
8π 3
0
(10)
According to the spin-fermion school of thought, the influence of the superconductivity on
the dynamical form factor is mostly due to a redistribution of spectral weight and explicit
calculations by Abanov and Chubukov (2000) demonstrate that ∆EJ is of order 10 K, in
fact already comparable to the condensation energy. As Demler and Zhang (1998) argued,
this sum rule acquires a more direct significance if the magnetic resonance corresponds with
74
a pure π mode. The normal state has to be viewed in the SO(5) language as a state
where the SO(5) superspin has disappeared due to amplitude fluctuations. Accordingly,
the spectral weight of the resonance has disappeared as well. Below the superconducting
transition antiferromagnetic correlations appear which were absent in the normal state, and
these manifest themselves through the weight of the magnetic resonance/π mode, emerging
‘from nowhere’. Hence, one should take the total weight of the magnetic resonance into
account in the sum rule eq. (10), and this leads to a δEJ ' 18 K which is again of the right
order.
Chakravarty and Kee (2000) arrived at a different sum rule which has a more fundamental status. Without invoking any assumption, they demonstrate that for a singlet d-wave
superconductor,
3
∆S(π/a, π/a) = λM
2
where ∆S(~q) is the change in the instantaneous structure factor S(~q) =
(11)
R∞
−∞
dωS(~q, ω) due
to the presence of the off-diagonal long range order, while λM is the condensate fraction
corresponding with the fraction of the electrons participating in the pairing. In a weak coupling BCS superconductor λM = N (0)∆, expected to be unobservable given the resolution
of neutron scattering. Assuming that ∆S equals the spectral weight of the resonance one
arrives instead at a condensate fraction of a couple of percents, an estimate which seems
quite reasonable for a high-Tc superconductor.
3.7. Concluding remarks.
In this section we have reviewed theories dealing with the magnetic resonance, and in
the process we have covered much of the theoretical main stream. The reader might have
perceived it as a bit confusing. It reflects the state of the field: the problem of high-T c
superconductivity is not solved. Instead, there is a patchwork of theoretical ideas, devised
to explain some aspects of the data successfully, failing badly in other regards. A large
75
majority of the theorists will agree that spins/magnetism are important, if not at the heart
of the problem. Next, the school of thought finding its inspiration in the idea that one can
at least start out with a conventional fermiology is quite influential. Indeed, only knowing
about the optimally doped superconductors at low temperatures, it looks like as if all the
work is done.
However, it appears to us that this is too easy. This apparent success involves a subtle
conceptual flaw. On closer inspection, the fermiology philosophy rests on the quite reasonable notion that at large lengths and long times the high-Tc superconductors behave quite
like conventional, Fermi-liquid derived BCS superconductors. This is not quite the case at
the short times and short lengths where the pairs are formed, as exemplified by the anomalous, non Fermi-liquid behaviors found at high temperatures in the normal state. It is then
implicitly taken for granted that one can get away with the free quasiparticles of the long
wavelength limit, getting increasingly ‘shaken and stirred’ by the effects of the interactions.
After a moment of thought, one should come the conclusion that this can’t quite work. On
the lattice scale there is no such thing as a free fermion, as should be clear to anybody
having familiarity with the t − J model. The spins and holes span up a Hilbert space of a
completely different nature than that of the Fermi-gas, and the dynamics of these spins and
holes is much richer (or ‘much more complicated’) than the empty and silent world of the
Fermi-gas. This is probably best illustrated with the considerations in the previous section
regarding stripe formation. The conclusion is that the scaling flow is irreversible. Starting
from a very complex physics at short distances matters apparently can simplify to something
as simple as the BCS superconductor, believing for the time being that the guess for the
fixed point is correct. However, this is only possible because information content gets lost,
and it is impossible to recover this information by starting at the low end, and trying to
climb upward on the energy ladder.
More than anything else, stripes symbolize this problem. We believe the basic understanding of the nature of static stripes as discussed in section 2 should be correct. However,
experiments suggests another entity called ‘dynamical stripes’. It is quite well established
76
in the 214 system, and it refers to experiments demonstrating that on rather short time
and length scales correlations exist in the cuprate electron system which are quite like those
found in static form when stripes are condensed. These will be discussed at various instances in the remainder of this text. However, despite these dynamical stripe complexities,
214 does not look all that different from the other superconductors at long length- and time
scales. It carries nodal fermions, it is thermodynamically quite similar, and T c can still be
quite high. At the same time, we already discussed the convincing evidence for strong stripe
correlations in heavily underdoped 123. We already draw attention to the fact (fig. 13) the
dynamical factor of this ‘stripy’ 123 does not look that different from those at higher dopings
which seem so effortlessly explained with RPA. Although weak, there is still a resonance, but
now the ‘incommensurate side branches’ have clearly to do with the stripes. Somehow, this
stripy affair transforms in a very smooth fashion into the RPA-like physics of the optimally
doped superconductor. Another disturbing fact is the one dimensional polarization of the
incommensurate peaks in the single domain 60 K 123 superconductor discovered by Mook
et al. (2000). Very strange things have to happen with Fermi-surfaces to make this possible
in RPA, while it is trivially explained with stripes, see Zaanen (2000a).
The theorists are in the unpleasant situation that nobody seems to have a clue how
to unify the BCS-ish features with the stripy side. Starting from the static stripes, nodal
fermions should not be present. The resonance is equally hard to explain. The only proposal
in this regard, Batista, Prtiz and Balatsky (2001), suggests that the resonance has to do
with a van Hove singularity in the stripe spin-wave spectrum, and this is clearly falsified by
the experiments on strongly underdoped 123. Starting from the weak coupling side, there is
little hope for finding back the slightest sign of a stripe correlation employing manageable
Feynman graphs. The conclusion is that the main result of 15 years of research in high-T c
superconductivity has made clear that the mystery is far deeper than initially expected.
77
(a)
ω
(π,π)
q
(0,π)
(b)
ω
(c)
ω
(π,π)
(π,π)
FIG. 13. Artist impression of the evolution of the magnetic dynamical formfactor of the
cuprates at low temperatures on the energy interval 0 < ω < 50 meV, and wavevectors ~k along the
(π/a, π/a) → (π/a, 0) direction in the vicinity of the (π/a, π/a) point. In La2−x Srx CuO4 (a) only
incommensurate fluctuations are found (hatched area’s) living at a wave vector which appears to
be identical to the ordering wave vectors of the static stripe antiferromagnets. It seems that the
main difference with the static stripes is that in the optimally doped superconductors (x ' 0.15,
Tc ' 40K) a small energy gap (∼ 5 meV) opens up at the low energy end of the triplet spectrum,
suggestive of fluctuating stripe order. The triplet spectrum of optimally doped YBa 2 Cu3 O6+y
with y ' 0.9 and Tc ' 90K is completely dominated by an excitation which is sharp in energy and
momentum: the ‘magnetic resonance, indicated by the black dot in (c). Upon reducing doping
somewhat (y ' 0.7, Tc ' 60K), ‘incommensurate’ side branches appear, (c), suggesting that the
resonance is part of a mode with an ‘inverted’ dispersion. This finds an appealing explanation in
the fermiology language of section 4.5 in terms of the Bogoliubov excitations of a BCS-like d-wave
superconductor. However, upon further reducing doping to y ' 0.37 the recent data by Mook,
Dai and Dogan (2002) indicate the onset of static stripe order. The spectrum is again dominated
by the incommensurate fluctuations of the 214 kind (b), although there is still a resonance discernible which seems to ‘interrupt’ the incommensurate fluctuations. To reconcile the stripe-like
incommensurate fermions with the magnetic resonance is an important open problem.
78
Temperature T / J
0.5
0.4
0.6
0.4
0.3
AF
∆0(x) / J
0.2
0
0.2
0
0.1
0.2
TBE
Td
0.1
AF
0
Tc
spin gap
0
d-wave SC
0.05
0.1
0.15
0.2
0.25
Hole filling x
FIG. 14. The ‘classic’ phase diagram of the high-Tc superconductors, as suggested by the
spin-charge separation theories, Brinckmann and Lee (2001). The electrons fractionalize in pure
spin 1/2 excitation (spinons) and charge −e excitations (holons). The holons are hard-core bosons
occurring at a density x, which are therefore Bose-condensing at a temperature T BE ∼ x. The
spinons behave like fermions subjected to attractive interactions in the d-wave channel, and they
undergo a BCS instability at Td , and this temperature is at maximum at half-filling to go down
with doping. Below Td a spin gap opens with d-wave symmetry and the doping dependence of the
size of this gap (∆0 (x)/J) is indicated in the inset. In addition, on the RPA level an instability
towards antiferromagnetism is found (dotted line), supposedly strongly weakened by higher order
corrections, such that only antiferromagnetism is found at very low dopings. Only when both
charge and spin are condensed one is dealing with a normal superconductor. Since T d > TBE in
the underdoped regime, a spin gap is opening up (going hand in hand with the appearance of the
resonance) at a temperature (much) higher than Tc : the explanation of the ‘pseudo-gap phase.’ It
is noticed that in the regime where the theory can be truly controlled (large-N ) the present phase
diagram is superseded by the phase diagram of Vojta and Sachdev (1999), see fig. 10.
79
0.5
0.3
ky
2π
0.1
-0.1
-0.3
-0.5
-0.5
-0.3
-0.1
0.1
0.3
k x / 2π
80
0.5
FIG. 15. Illustration of the ‘dynamical nesting’ condition giving rise to the singularities in
the Lindhard function of the d-wave superconductor, responsible for the binding of the resonance
(Brinckmann and Lee 2001). The square indicates the magnetic Umklapp surface (the arrows
correspond with ~q = (π/a, π/a)). At half-filling the nodes of the d-wave Bogoliubov quasiparticles
lie on the Umklapp surface according to the spin-charge separation scenario and the Lindhard
function has weight all the way to zero-energy. On the RPA level, the resonance ‘hits the zero
of energy’ causing an antiferromagnetic instability. However, upon doping the nodal points move
towards the center of the Brillioun (the dots in the figure). One should now imagine what happens
when one could at will raise the Fermi-energy: this energy cost corresponds with the minimal
energy needed to find particle-hole excitations. This ‘dynamical Fermi surface’ corresponds with
the ellipses in the figure centered around the nodal points. At the moment that they hit the
Fermi-surface χ000 becomes non-zero. In the inset the energies of the particle-hole excitations are
shown in the upper right corner of the Brillioun zone, with the minima ω0 (‘hot spots’) as dots
while the contours refer to higher energies. It is noticed that due to a special choice of the bare
dispersions the nodal ‘ellipses’ turn into ‘banana’s’ with a flat outer face being parallel to the (π, π)
Umklapp surface. This strengthens the singularity in χ0 significantly and Brinckmann and Lee
argue that this dynamical nesting effect is required for the binding of the resonance.
81
1.0
1.1
1.2
1.3
1.4
2
Amplitude / (µB /J)
30
20
10
0
0.1
20
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
10
0
−0.2 −0.1
1.5
0
0.1
0.2 −0.2 −0.1
0.1
0.2 −0.2 −0.1
0.5
0.4
0.3
0.2
2
Amplitude / (µB /J)
0.2 −0.2 −0.1
1.0
0.95
0.9
0.8
0.7
2
Amplitude / (µB /J)
0
−0.2 −0.1
30
1
0.5
0
−0.2 −0.1
0
δ / 2π
δ / 2π
FIG. 16. The imaginary part of the full dynamical magnetic susceptibility χ00 (~q, ω) as calculated
by Brinckmann and Lee (2001). These are plotted as function of wave vector in the two directions
as indicated around the (π/a, π/a) point. Top row: sequence of energies ω/ω res = 1.0 − 1.4, at and
above the resonance; the response broadens out rapidly at energies above the resonance because
the featureless particle-hole continuum of the metal takes over. Middle row: ω/ω res = 1.0 − 0.7;
the incommensurate side branches become visible. Bottom row: energies far below ω res ; the signal
becomes very weak (notice change of scale), and notice the flip from the ‘horizontal’ to the ‘diagonal’
orientation of the incommensurate peaks.
82
4. Experimental techniques
Thermodynamic methods like susceptibility or magnetization measurements are well
known to yield reliable information about bulk magnetism, while specific heat data are essential for the analysis of the entropy content in (magnetic) phase transitions. Refinements
of the possible spin structures require neutron scattering, µSR, and magnetic resonance
methods. In stripe systems, where both charges and spins are involved, angular resolved
photoemission spectroscopy (ARPES), scanning tunneling microscopy (STM) and X-ray
play equally important roles. Below we illustrate the merits of these techniques by recent
examples. We will start by a more extensive explanation of NMR/NQR as the technique
is sensitive for both charges and spin and is complementary to those of neutrons regarding
time scales. Various aspects of NMR/NQR are also important for µSR. In HMM10 the
reader will find a similar concise treatise of neutron scattering.
4.1. NMR
Nuclear relaxation rates, line positions, linewidths and line intensities are the main
NMR/NQR parameters, see Abragam (1961), Winter (1970), Slichter (1991), and van der
Klink and Brom (2000). For the application to high-Tc ’s various introductions and reviews
have appeared. Of the more recent ones we mention those of Mehring (1992), Asayama et
al. (1996), Berthier et al. (1996), and Brom (1998). Here we update these reviews with
emphasis on the work on stripes.
4.1.1. The NMR method.
A standard technique in pulse NMR is the echo method. By application of a π/2 pulse
the nuclear magnetization component along the direction of the static field B0 (the z-axis) is
rotated to the xy plane ⊥ B0 . After a delay time τ a second (π) pulse, twice the duration of
83
the π/2 pulse, is applied that refocusses the magnetization in the xy-plane after a time 2τ .
The height of the echo is then proportional to the magnetization originally along the z-axis,
Mz . By slightly different procedures two important relaxation rates can be measured. The
measuring cycle of the nuclear Zeeman relaxation rate (T1−1 ), which is a measure for the
rate of energy exchange of the nuclear spins with the surrounding, usually starts by a comb
of π/2-pulses, by which Mz is destroyed. Thereafter one measures the recovery of Mz via
the echo-method. Study of the nuclear relaxation rate is often a convenient method to find
details about the electron spin dynamics. In a determination of the transverse or spin-spin
relaxation rate (T2−1 ) the memory loss in the refocussing process in the xy plane is measured
by varying the delay time τ in the echo method. Every cycle starts with the same rotation
of Mz to the xy-plane. T2 is a measure for the time correlation of the local field fluctuations,
which is often determined by the interaction between the nuclear spins. If the echo in the
time domain is Fourier transformed to the frequency domain, the position and width of the
resonance curve can be analyzed.
Knight shift and line width. In general the relative shift of the resonance line with respect
to a suitable reference, K = ∆B/B, contains an orbital (Korb ), a diamagnetic (Kdia ) and a
spin (Kspin ) contribution: K = Korb + Kdia + Kspin . In simple s-electron metals the hyperfine
interaction between a nuclear spin (I) and electronic spins (S) Hhf =
P
i
I · Ai · Si is often
dominated by the scalar Fermi contact term A/γn γe h̄2 ≈ (8π/3) < |uk (0)|2 >EF , with
< |uk (0)|2 >EF the density of electrons at the Fermi energy at the nuclear site (Korringa
1950, Slichter 1991). The spin contribution (usually called Knight shift) is then given by
Kspin =
∆B
A<S>
A
=
=
χ0 ,
B
γn h̄
gµB h̄γn
(12)
with χ0 the spin susceptibility (in SI units χ0 = χmolar /NA µ0 with NA Avogadro’s number),
g = 2 the Landé g-factor, µB = γe h̄/g the Bohr magneton, and A the already introduced
scalar Fermi contact interaction term. The electron spin gyromagnetic ratio γ e is typically a factor 103 larger than the nuclear spin gyromagnetic ratio γn . One contribution to
84
the linewidth stems from the T2 processes mentioned above and is called the homogeneous
broadening. A distribution of static (on the time scale of the experiment) magnetic fields
gives an inhomogeneous contribution.
Spin-lattice relaxation. Fluctuating magnetic fields with a frequency component (spectral
density) at the frequency of the nuclear Zeeman splittings (in an applied magnetic field
B) cause transitions between the levels involved. Often the spectral density originates in
electronic spins (S) linked to the lattice (the thermal reservoir). The transition probabilities
W that determine the Zeeman relaxation rates of the various nuclear spins α I are then
functions of the imaginary part χ00 of the electronic susceptibility χ. For B k z, α Wz is given
by (Zha et al. 1996, Moriya 1963):
α
Wz =
kB T X α
| F⊥ (q)|2 χ⊥ 00 (q, ω → 0)
4h̄2 µ2B ω q
(13)
In this equation q denotes the electron wave vector, µB the Bohr magneton. The real and
imaginary parts of the electronic susceptibility are given by χ0 and χ00 . The form factors
F⊥ (q) in eq. (13) denote the components of the spatial Fourier transforms of the hyperfine tensor A in the plane perpendicular to the z-axis. The hyperfine interaction between
the nuclear spin (I) and an electronic spin (S) at a particular site n is now written as
Hhf = Axx Inx Snx + Ayy Iny Sny + Azz Inz Snz . For spins I = 1/2, T1−1 equals 2W . For I > 1/2 63
Cu and
65
Cu have I = 3/2 and
17
O has a spin I = 5/2 - electric field gradients at the
nucleus give a splitting in zero magnetic field. If the zero-field splittings are large enough
(for
63
Cu in the high-Tc compounds they are of the order of 20 - 30 MHz) the relaxation
rates and Knight shifts can be determined not only by NMR but also by zero-field NQR, see
e.g. MacLaughlin (1976). The (in general non-exponential) recoveries measured by NQR
and NMR are related to the same transition probability W , Narath (1967).
Spin-spin relaxation. The signal induced in a coil due to the rotating in-plane nuclear
spin components, decays due to spin dephasing T2 and T1 processes according to (Slichter
85
1991):
I(t) ∝ M (2t) = M (0) exp(
2t2
−2t
− 2 )
T2R
T2G
(14)
−1
where T2R
= (2 + R)/(3T1 ) is the Redfield term and T2G is the Gaussian part of the echo
decay assuming static neighbors. Here, R is the T1 anisotropy ratio, measured in and out of
the plane. For the Cu-sites in the high-Tc compounds with B k c the Gaussian component
of the transverse relaxation rate T2G takes the form (Zha et al. 1996):
−2
T2G
=
0.69
128h̄2 µ4B

"
#2 

1 X
X
1
eff
4 0
2
eff
2 0
,
F
(q)
[χ
(q,
0)]
−
F
(q)
χ
(q,
0)
z
z
z
z

N q
N q
(15)
where Fzeff are effective form factors. In such cases T2 is a probe for χ0 (q, 0). The transverse
relaxation or echo decay time T2 is sensitive for motions that are typically a few orders of
magnitude slower than seen in T1 , and the field fluctuations that cause the relaxation are
along the external field direction. In solids in the absence of conduction electrons, T 2 is often
determined by the direct nuclear dipole-dipole interaction between like and unlike nuclear
spins, see Abragam (1961), Slichter (1991), Walstedt and Cheong (1995). In the presence
of conduction electrons the nuclear interaction can be enhanced by the polarization of the
electrons.
NMR-NQR signal intensity. In a standard NMR-NQR experiment the amplitude of the
detected voltage in the coil is proportional to the nuclear magnetization and depends on the
waiting time τ after the π/2 pulse as:
I ∝ M (2τ ) = M0 exp(−
2τ
N γ 2 I(I + 1)Bef f
) M0 =
T2
3kB T
(16)
The extrapolated M (2τ = 0) = M0 , integrated over the frequency window and corrected for
T (Curie law) and T2 effects has to be temperature independent.
4.1.2. Metals, superconductors and magnetic materials
86
In the CuO2 layered compounds, like doped La2 CuO4 , magnetic, metallic, and superconducting features are present. Below we look in more detail into these particular cases.
4.1.2.a.
Magnetic Materials. Let us start by illustrating the power of NMR-NQR
to study antiferromagnetism with the energy levels of
139
La in La2 CuO4 .
a nuclear spin I = 7/2 and a nuclear quadrupole moment.
139
La has
The line position and
shape of the NMR-NQR spectrum are determined by the following hamiltonian: Ĥ =
−−i→ P
P
i
i ˆ2
i
i
ˆ2
ˆ2
ˆ2
− i h̄γi Iˆi .Bef
i [h̄vQ /6](3Iiz − Ii + η (Iix − Iiy )) where vQ = [eQi Vzz ]/[4h̄Ii (2Ii − 1)]
f +
i
i
]/[Vzzi ]. The index i runs over all nuclear species with spin Iˆi , a nuclear
− Vyy
and η i = [Vxx
i
gyromagnetic ratio γi and an electric quadrupole moment Qi . Bef
f is the effective magnetic
i
field at the nuclear site i that determines the magnetic properties of the spectrum. V zzi , Vxx
i
and Vyy
are the electric field gradient (EFG) tensor components at the nuclear site i. At
zero field and η = 0, the hamiltonian is simplified to Ĥ =
P
i
ˆ2
i h̄vQ (3Iiz
− Iˆi2 )/6 and for
vQ = 6.0 MHz leads to three distinct resonance frequencies at 6, 12, 18 MHz respectively.
When a static local magnetic field is present at the
139
La nuclear site, the full hamiltonian
has to be considered. Diagonalization of the hamiltonian gives a splitting for each of the
three lines into more than one line as a function of the growing magnetic field. The magnitude of the splitting depends on the magnetic field value and its orientation with respect
to Vzz . Fig. 17 shows the splitting as a function of the development of internal field in the
x − y plane, see also MacLaughlin (1994), and Borsa et al. (1995). Matsumura et al. (1997)
used
63
Cu(2) NQR to study the nature of the ordered two-dimensional antiferromagnetic
spins in this same compound.
Relaxation rates. The dynamics of the 2 dimensional quantum Heisenberg antiferromagnet (2DQHAF) is conventionally probed by neutron scattering (NS) and NMR-NQR
techniques. Let us consider here how to make use of the relaxation times discussed in
the Appendix to extract information about the dynamic properties of 2DQHAF. When
−
scaling conditions are used, the generalized susceptibility has the form χ00 (→
q , ωR ) =
87
S(S + 1)ξ z f (qξ, ω/ξ z )/(3kB T ) (Rigamonti et al. 1998), where ξ is the magnetic correlation length and z is the dynamical scaling exponent which depends on the properties
The scaling function f is given by f (qξ) = β/1 + q 2 ξ 2 where
of the spin dynamics.
−
1/β = [ξ 2 /(4π 2 )] d→
q /(1 + q 2 ξ 2 ) is a normalization factor that has been introduced to preR
serve the sum rule
P ¯¯ α ¯¯2
q ¯Sq ¯ = ²N S(S+1)/3. ² is the reduction factor for the spin fluctuations
amplitude in a 2DQHAF, Chakravarty et al. (1989). The correlation length ξ can be related
√
to 1/T1 (see eq. 13). For 63 Cu in La2 CuO4 it reduces to 1/T1 ≈ 4200(ξ/a)z /[ln(2 πξ/a)]2 .
The value of ξ(T) extracted from the 1/T1
63
Cu NQR data in La2 CuO4 (Caretta et al. 1999)
can be compared to the theoretical predictions of Chakravarty et al. (1989), and Hasenfratz
and Niedermayer (1991) for the renormalized classical regime. The fitting is excellent when
z = 1 is used and is in full agreement with the correlation lengths extracted from neutron
scattering.
Furthermore, the correlation length can be related to the Gaussian part of the spin-spin
relaxation time 1/T2G (Rigamonti et al. 1998, Caretta et al. 1999, Berthier et al. 1996). In
the quantum critical regime (z = 1 and ξ >> a), T1 T /T2G is temperature independent.
Signal intensity. Close to the ordering temperature the magnetic fluctuations slow down
that much that detection of the nuclei by NMR is no longer possible. Especially due to
the work on the 214 cuprates and nickelates, it has become clear that the region where
the signal starts to disappear (the so called wipe-out regime) gives detailed information
about the spin dynamics, which is otherwise almost impossible to get. In a pulsed NMRNQR experiment, the total signal intensity is proportional to the nuclear magnetization M
given by: M (2t) = M0
R∞
0
[exp(−2t/T2 (ω))]F (ω)dω, where F (ω) is a normalized frequency
distribution function which determines the line shape. t is a waiting time after the π/2
pulse, which is usually set to be t = td as a minimum experimental ”dead” time. M (0)T
should be temperature independent, as the nuclear magnetization follows the Curie law.
This basic NMR concept seems to be violated in strongly underdoped high-Tc cuprates,
where the signal is often lost far above the magnetic freezing or ordering temperature. A
88
description for the experimental observation of this wipe-out effect is discussed below. Curro
et al. (2000), Suh et al. (2000) and Teitel’baum et al. (2000) showed that wipe-out effects
in the cuprates are due to a distribution of short dephasing times (caused by a distribution
in correlation times (τ ) or activation energies, because τ = τ∞ exp(Ea /T ). The distribution
of the activation energies Ea is not known beforehand and might be e.g. half Gaussian or
Gaussian. Due to the distribution in activation energies M becomes:
M (2t) Z ∞ Z ∞
=
exp(−2t/T2 (ω, Ea ))F (ω)dωP (Ea )dEa .
N M0
0
0
(17)
If the signal is governed by various magnetization recovery rates, the resulting time
dependence can often be approximated by a stretched exponential of the form M (0) ∝
exp[−(2t/T2 )α ]. In the usual echo pulse sequence π/2-t-π-t, t can never be smaller than the
dead time td of the set-up. Fig. 18a gives a numerical plot for the signal decay at three
temperatures as a function of 2t. It shows that an extrapolation of M (t) to t = 0 with the
apparent T2 will always be less than M0 due to the finite dead time.
In case of a Gaussian distribution of activation energies, the signal extrapolated to time
zero, is given by Teitel’baum et al. (2000):
Z E2
M (0, ω)
−2td
Ea − E 0 2
exp(
) exp(−( √
=
) )dEa +
2td
T2 (ω, ∆)
N M0 exp( T2 (ω) )
E0
2∆
Z
∞
E1
exp(
Ea − E 0 2
−2td
) exp(−( √
) )dEa
T2 (ω, ∆)
2∆
(18)
The first term in eq. (18) represents the signal decay, while the second term is responsible
for the recovery. E1 is related to E2 by E2 − E1 = T ln(τ2 /τ1 ), where the cut-off τi follows
from the hyperfine coupling and the shortest measurable dephasing time (Teitel’baum et al.
2000). A typical numerical solution for eq. (18) is shown in fig. 18b. In real systems the
spin dynamics changes when entering the low temperature phase and the low temperature
recovery cannot be described by the high temperature physics, see sections 5 and 6.
4.1.2.b. Strongly correlated metals. In simple metals, where the Fermi contact term of
the s-electrons dominates, the susceptibility is of the Lindhard form (Shastry and Abrahams
89
1994):
χ0L (q, ω)
= χP
"
1 q
1−
3 2kF
µ
¶2 #
, χ00L (q, ω) =
πχP ω
θ(|ω − vF q|)
2 vF q
(19)
to leading order in q, ω (the spin susceptibility χ0 in eq. (12) now equals χ0L (q, 0)). Here
χP is the Pauli susceptibility, θ the step function and vF the Fermi velocity (Ashcroft and
Mermin 1976). Eq.(19) leads to the well-known Korringa relation (Korringa 1950),
1
K T1 T =
πh̄kB
2
Ã
µB
γn
!2
=S
(20)
with S the Korringa constant. This equation links the relative Knight shift (K) to the
nuclear-spin lattice relaxation rate (T1−1 ). Although different independent electronic contributions to the Knight shift might cancel each other, they always add in the relaxation rate.
In the presence of an effective on-site repulsive Coulomb interaction U the susceptibility will
be Stoner enhanced (Moriya et al. 1963, Shastry and Abrahams 1994, van der Klink and
Brom 2000) and the simple Korringa relation has to be modified. Below we give more details
about the NMR approach in the strongly correlated cuprates. We use the 123-compounds,
see section 8, as example.
The Mila-Rice-Shastry hamiltonian. Mila and Rice (1989), see also Shastry (1989), derived the effective electron-nuclear interaction for the various nuclei in YBa 2 Cu3 O7−δ . The
nuclear spin of
63
Cu at the lattice site ri (63 I(ri )) appears to interact not only with the
electron spin at ri but to have also a non-negligible transferred hyperfine coupling to its four
neighboring Cu2+ spins:
Hhf =
X
α
63

I α (ri ) 
X
Aα,β Sβ (ri ) + B
NN
X
j
β

(21)
63
Cu nuclei to the Cu2+
Sα (rj )
where Aα,β is the tensor for the direct, on-site coupling of the
spins, and B is the strength of the transferred hyperfine coupling. Similar expressions can
be written down for the hyperfine interactions of the nuclear spins of
(89 I) with Cα,β resp. Dα,β the transferred hyperfine couplings of the
90
17
17
O (17 I) and
O resp.
89
89
Y
Y nuclear
0
0
spins to its nearest-neighbor (NN) Cu2+ spins, and Cα,β
resp. Dα,β
its coupling to the next-
nearest-neighbor (NNN) Cu2+ spins (C and D are scalars in the SMR form). The C 0 -term
is not present in the original SMR hamiltonian and is introduced by Zha et al. (1996).
Knight shift. The expression for the Cu(2) Knight shift contains Ac , Aab and B as three
free parameters (the static spin susceptibility χ0 is in principle known from experiment). As
the experiment gives only the Knight shift in the c and ab direction, Kc and Kab resp.
63
Kc =
(Ac + 4B)χ0
,
γe 63 γn h̄2
63
K ab =
there remains one parameter undetermined.
(Aab + 4B)χ0
,
γe 63 γ n h̄2
(22)
The unknown parameter is often chosen
to be the ratio αr between Aab and 4B (Millis, Monien and Pines 1990; Millis and
Monien 1992).
17
For the oxygen sites in the CuO2 -plane the Knight shift is given by
K β = [2(Cβ + Cβ0 )χ0 /γe 17 γ n h̄2 . For yttrium a similar equation holds. The description
of the Knight shift in La2−x Srx CuO4 and YBa2 Cu3 O7−δ for various values of x or δ with the
same parameter set works well and is a strong point for the one spin model, see Millis et al.
(1990), Alloul et al. (1988), Monien et al. (1991), Takigawa et al. (1991).
In underdoped compounds χ0 decreases below some temperature T0 well above Tc , see fig.
3, which can be seen as the opening of a pseudogap (see pseudogap) (Pines 1997, Berthier
1996).
Relaxation rates. In contrast to the Knight shifts the T -dependencies of the relaxation
rates for the various nuclei differ considerably. For the Cu nuclei 1/T1 T has a maximum
at (in case of optimal doping) or above (for underdoped samples) Tc , and follows often a
Curie-Weiss like behavior (Asayama et al. 1996). For oxygen and yttrium T1 T is Korringalike (due to the inaccuracy in K it is experimentally hard to distinguish whether T 1 T K or
T1 T K 2 is constant, see Berthier et al. 1996).
Enhancement of the dynamic susceptibility due to antiferromagnetic (AF) fluctuations
in combination with filtering due to the form factors can explain these results. This idea
91
is behind the nearly antiferromagnetic Fermi liquid (NAFL) model, proposed by Millis,
Monien and Pines (MMP) (1990). A normal Fermi-liquid and an antiferromagnetic part
(nearly antiferromagnetic fermi liquid) make up the susceptibility:
χ(q, ω) = χFL (q, ω) + χNAFL (q, ω)
(23)
The Fermi liquid like dynamic susceptibility has the form χFL (q, ω) = χq /(1 − iω/Γq ). Γq is
comparable to the bandwidth and χq ∼ χ0 . The real and imaginary part of χFL are χ00FL =
ωπχ0 h̄/Γ and χ0FL = χ0 . The antiferromagnetic part dominates close to Q = (π/a, π/a),
and has a Lorentzian dependence on the correlation length ξ: χNAFL = χQ /[1 + (Q − q)2 ξ 2 −
iω/ωSF ] with the static susceptibility at the antiferromagnetic wave vector χ Q ∝ ξ 2 . MMP
propose that the frequency of the relaxational mode, ωSF , depends on ξ as ωSF ∝ ξ −2 .
Adding χFL and χNAFL gives (for low ω)
πχ0 h̄ω
(ξ/a)4
χ00 (q, ω) =
1+β
Γ
[1 + (q − Q)2 ξ 2 ]2
#
"
(ξ/a)2
0
χ (q, ω) = χ0 1 + β
1 + (q − Q)2 ξ 2
"
#
(24)
The real and imaginary part of the susceptibilities are linked via the Kramers-Kronig relations. Apart from the magnetic correlation length ξ(T ) and the broadening Γ the proposed
form for χ has as an additional parameter β, which gives the relative strength of the AF
fluctuations to the zone center fluctuations. The form factors F (q), see eq. 13, follow from
the hyperfine interaction hamiltonian (the hyperfine fields act coherently, see Berthier et al.
1996; Millis, Monien and Pines 1990). For sufficiently large values of ξ (ξ/a > 2),
63
(T1 T )−1
varies as (ξ/a)2 , while 1/63 T2G ∝ (ξ/a), see eq. (15).
4.1.2.c. Superconductors. As superconductivity means shielding of applied magnetic
fields, at first sight it seems surprising that NMR remains possible in the superconducting
state. In type I superconductors conventional NMR experiments are indeed hampered in
their sensitivity for three reasons: (1) the applied static magnetic field is shielded, which
means that only the nuclei within the penetration depth λ feel a (very inhomogeneous)
92
static field, (2) the rf-pulses cannot penetrate the sample except for this outer layer, and (3)
the magnetization of the nuclei which leads to the final NMR signal is shielded again. In
type II superconductors the situation is different as the external field penetrates the sample
via vortices and gives a more homogeneous field distribution. Also the rf pulses are not so
strongly shielded from the interior as they bend the vortex pattern. NMR on powders and
single crystals with the right geometry is possible, albeit the line intensity is much reduced
with respect to the normal state (Moonen and Brom 1995).
Due to spin pairing the spin susceptibility χ0 and hence the Knight shift (K ∝
χ0 ) is expected to decrease with T according to the Yoshida function:
µ2B
R∞
∆
K
∝
NBCS (E)[∂f (E)/∂E]dE with the density of excited states NBCS (E) = N (0)|E|/(E 2 −
∆2 )1/2 . The opening of the superconducting gap increases the density of excited states just
above the gap (Schrieffer 1988, Tinkham 1975), while also the coherence factors will be
important (MacLaughlin 1976). As a result the nuclear relaxation can show the so-called
Hebel-Slichter enhancement below Tc . In fields close to Bc2 , the vortices are closely packed
and the nuclei in the ‘normal’ vortex cores can form an additional relaxation channel (Weger
1972).
4.2. µSR
In neutron spectroscopy the complex susceptibility χ(q, ω) can be measured as a function
of q, where NMR is only sensitive for certain q-values (depending on the location of the
NMR nuclei). The energy ranges of neutrons and NMR are almost complementary, neutrons
covering typically the range from 0.1 meV up to eVs, while the fluctuations measured by
NMR typically extend from 1011 s−1 to orders of magnitude below 1 Hz. By extrapolation
NMR and neutron data can be compared (Berthier et al. 1996, RossatMignod et al. 1991a
and 1991b).
The dephasing on the initial aligned moments of the muons is governed by the same fluc93
tuating fields that determine the NMR T1 . The muon-data (counts of positrons being the
decay product of the muon, which has a life time of 2.2 µs; the positrons are preferentially
emitted along the direction of the muon spin before decaying) are usually taken in low fields
(< 0.1 T) compared to NMR (> 0.1 T), and are sensitive for spin fluctuation rates in the
range 104 − 1011 s−1 . µSR (Brewer 1995) and NMR are also both sensitive for local magnetic
order. Representative zero-field µSR spectra for La1.8−x Eu0.2 Srx CuO4 are shown in fig. 19.
Klauss et al. (2000) performed µSR measurements for 0 ≤ x ≤ 0.2 and found a striking
nonmonotonic Sr concentration dependence for the static magnetic order. In the data for
x = 0 and x = 0.014 a clear precession of the time evolution of the muon spin polarization
is visible which proves the presence of a magnetic field at the muon site. This precession is
absent for higher doping 0.02 ≤ x ≤ 0.08. For x = 0.02 a strong decay of the muon spin
polarization signals the proximity of spin freezing, whereas there is no indication for static
magnetic order at 10 K for x = 0.04 and x = 0.08. However, clear signatures of static magnetic order reappear for higher Sr concentrations. A strong decay and a precession is visible
in the spectrum for x = 0.1 and the signatures of magnetic order are even more pronounced
for x = 0.12 and x = 0.15. Further increasing the doping suppresses the magnetic order
and for x = 0.2 the authors find only an onset of spin freezing at T = 8 K. For further
discussions see the section about 214-compounds.
4.3. Neutron scattering
Elastic and inelastic neutron scattering have played and still play an essential role in our
understanding of the normal and superconducting properties of the high-Tc cuprates. Due
to its spin sensitivity detailed information about static and dynamical stripes have been
obtained. HMM-10 gives a concise review of the relevant scattering theory. In fig. 20 we
reproduce the data of Mook et al. (2000) on YBa2 Cu3 O6.6 , showing the one dimensional
nature of the magnetic fluctuations in underdoped 123-compounds. The inelastic neutron
data show a four-fold pattern of incommensurate points around the magnetic (1/2,1/2) re94
ciprocal lattice position. The interpretation of such measurements had been unclear, with
both striped phases and nested Fermi surfaces being possible explanations. Because the
stripe phase is expected to propagate along a single direction, only a single set of satellites around the antiferromagnetic position should be observed. For La1.48 Sr0.12 Nd0.4 CuO4
Tranquada et al. (1995) suggested that the four-fold symmetry might arise from stripes
that alternate in directions as the planes are stacked along the c-axis, in accordance with
the tilt of the octahedra. In the 123-system, the a and b directions are inequivalent in the
CuO2 planes, due to the presence of the chains (orthorhombic structure), and it is expected
that in a single orthorhombic domain stripes preferentially align along one of the chains.
Mook et al. (2000) took scans of the magnetic incommensurate scattering for the twinned
and partially detwinned sample at 10 K. The scan centers of scan 1 is at (0.55, 0.55, 2)
in reciprocal lattice units (rlu), while the center of scan 2 is (0.45, 0.45, 2) rlu, both along
the b∗ -direction, connecting the IC peaks. Before detwinning the intensity ratio’s of the
satellites along the a∗ and b∗ direction are 1.05 ± 0.05. After detwinning 1.95 ± 0.21, which
is the same as the population of the domains for the two directions. The results support
one dimensional striped excitations.
4.4. ARPES
In angular resolved photoemission spectroscopy (ARPES) photons are absorbed by the
material and electrons are ejected. The control parameters are the frequency and the polarization of the incident photons and the measured quantities are the kinetic energy and the
angle of emergence (θ, φ) of the outgoing electrons relative to the sample normal. During the
high-Tc era the wave vector and energy resolution of the ARPES technique has improved by
orders of magnitude, making the technique one of the major sources of insight. In ARPES
the component of k parallel to the surface is conserved as the electron emerges from the
sample. In the quasi-two-dimensional cuprates this allows an unambiguous determination
of the momentum of the hole state from the measurement of the photoelectron momentum.
95
The most accurate ARPES data come from Bi2212, where the natural cleavage plane lies in
between the BiO bilayer, which results in extremely smooth surfaces with minimal charge
transfer. These conditions are crucial for ARPES, since it is a surface sensitive technique
due to the short escape depth of about 1 nm of the outgoing electron.
One main difficulty in precisely determining the Fermi-level crossing or the Fermi wave
vector (kF ) by ARPES lies in the fact that ARPES probes only the occupied states. This
procedure works well when the band dispersion is experimentally well defined. In more
difficult cases one might plot the ARPES intensity around EF as function of k - however
it is important to account well for the influence of the matrix elements (Kipp et al. 1999,
Bansil and Lindroos 1999, Borisenko et al. 2000 and 2001, Fretwell et al. 2000, Lindroos et
al. 2002).
In fig. 21 the one-dimensional electronic structure in La1.28 Nd0.6 Sr0.12 CuO4 is illustrated,
according to the ARPES data of Zhou et al. (1999). The dotted lines define the regions,
where the spectral weight is mainly concentrated. The Fermi surface expected from two
perpendicular 1D stripe domains in the 1D interpretation form a cross, that resembles the
dotted lines closely. Γ is the center of the Brillouin zone.
4.5. STM
Recently scanning tunneling microscopy (STM) has been used to image the electronic
structure of Bi2212 and revealed stunning results. It appears that in optimally doped Bi2212
the vortex cores are surrounded by a four unit cell periodic pattern of quasi-particle states,
see Hoffman et al. (2002a) and Sachdev and Zhang (2002), which is related to a localized
spin modulation. In underdoped Bi2212 even without a magnetic field STM reveals an
apparent segregation of the electronic structure into superconducting domains of a few nm
in size, located in an electronically distinct background, see Lang et al. (2002) and Zaanen
(2002). We discuss the experiment of Hoffman et al. (2002a) in more detail.
First the surface is imaged in the usual topographic mode, where the tip current is kept
96
constant by adjusting the height of the tip above the surface. The displacements of the
tip are translated to the structure of the scanned surface. Hoffman et al. (2002a) used in
addition a recently developed technique to measure the low-energy quasi-particle density at
the vortices (Pan et al. 2000) by taking differential tunneling conductance spectra as function
of sample bias voltage. Making a Fourier transform gives the charge density, n(q, ω), at the
measured energy interval as function of q.
Before applying the magnetic field, first the differential tunneling resistance (G = dI/dV )
is measured in zero-field. Because G(V ) is proportional to the local density of states (LDOS)
at E = eV , this results in a two dimensional LDOS map. Thereafter a similar scan is made
with the magnetic field (B). The two spectra, integrated over all additional spectral density
induced by the B field between the energies E1 and E2 at each location x, y are subtracted:
S(E2 , E1 )(x, y, B) =
P E2
E1 [LDOS(E, x, y, B)
− LDOS(E, x, h, 0)]dE with E1 = ±1 V and
E2 = ±12 V. A schematic image of the two-dimensional Fourier transform of the power
spectrum in 5 T is shown in fig. 22. Scans as function of the energy interval show that the
data are linked to the strong inhomogeneity in the samples, which leads to scattering events
that are sensitive to the peculiar gap structure.
4.6. Thermodynamic techniques, like susceptibility and heat capacity
Although the principles of thermodynamic techniques are known for a long time, the application to high-Tc compounds is far from trivial. In the specific heat data precise correction
for the lattice background appeared to be crucial (Loram et al. 1993), and in the analysis
the role of fluctuations appears to be a source of confusion, see van der Marel et al. (2002).
In the susceptibility data often only by taking derivatives relevant structures appear. In fig.
23 we sketch the susceptibility data of Lee and Cheong (1997) on La5/3 Sr1/3 NiO4 , where the
effects of charge and spin order are made visible by taking the T derivative d(χT )/dT of
χT . In samples with the same nominal composition, these effects are often less pronounced.
In this particular case there are three different phases denoted by I, II, and III with decreas97
ing T . Spin order occurs in phases II and III, but not in I. The anomalies establish that
the charge-spin correlations in the three phases are static on a time scale more than a few
seconds. Charge correlations are much shorter in I than in II or III.
98
12.0
11.9
ω (MHz)
11.8
7
6
5
0.00
0.05
0.10
Field (T)
FIG. 17. Transition frequencies for the first two zero-field transitions, if quadrupole interaction
and Zeeman energies (e.g. due to the development of an internal field) are both relevant. The
illustration is made for
139 La
with I = 7/2. It is assumed that
139 La
has zero field splittings of 6.0,
12.0, and 18.0 MHz. These values are close to those typically found in the cuprates and nickelates.
The field splitting depends on the angles θ and φ, which fix the orientation of the magnetic field
with respect to the electrical field gradient EF G. Here the internal field is assumed to develop in
the x − y plane - i.e. the angle θ with the main component of the electrical field gradient (V zz )
equals π/2 - and φ = 0. The anisotropy parameter η is taken to be η = 0.02. The 18.0 MHz
transition is not shown, after Issa Abu-Shiekah (2001).
99
FIG. 18. (a) Numerical solution of the time dependence of the signal intensity I according to
eq. (17) at f =38.2 MHz at 120 K (drawn), 60 K (dotted) and 30 K (dashed). Due to the rf pulse
at t the echo appears at time 2t. The parameters ∆ and E0 are 28, 60 K. Iex is the value at t = 0,
extrapolated from the decay after the dead time 2td . The figure shows that an extrapolation of the
magnetization data M (t) to time zero (t = 0) with the apparent T2 will always be less than the
real magnetization due to the non-zero dead time of the set-up (the magnetization M and intensity
I are proportional to each other). The higher the spin dephasing rate the larger this error will be.
(b) Numerical solution for eq. (18) with the same parameters as in a, E2 /T = 1 and E1 /T = 8,
after Abu-Shiekah (2001). The reappearance at low temperatures has the same explanation as the
wipe-out at high temperature. The slowing down of the electron spin fluctuations lead to very
short dephasing times in a certain temperature window. At high T the fluctuations are too fast,
at low T too slow. In real systems the spin dynamics changes when entering the low temperature
phase and the low temperature recovery cannot be described by the high temperature physics, see
e.g. next section.
100
0.25
a
asymmetry
asymmetry
0.15
0.10
x=0
T = 10 K
0.05
0.00
0.0
0.5
1.0
time µs
1.5
2.0
b
0.20
0.15
0.10
0.0
x = 0.08
T = 10 K
0.5
1.0
time µs
1.5
2.0
FIG. 19. Representative zero-field µSR spectra at T = 10 K for La 1.8−x Eu0.2 Srx CuO4 , after
Klauss et al. (2000). Klauss et al. performed µSR measurements for 0 ≤ x ≤ 0.2 and found
a striking nonmonotonic Sr concentration dependence for the static magnetic order. In the data
for x = 0 and x = 0.014 a clear precession of the time evolution of the muon spin polarization is
visible which proves the presence of a magnetic field at the muon site. This precession is absent
for higher doping 0.02 ≤ x ≤ 0.08. For x = 0.02 a strong decay of the muon spin polarization
signals the proximity of spin freezing, whereas there is no indication for static magnetic order at
10 K for x = 0.04 and x = 0.08. However, clear signatures of static magnetic order reappear for
higher Sr concentrations. A strong decay and a precession is visible in the spectrum for x = 0.1
and the signatures of magnetic order are even more pronounced for x = 0.12 and x = 0.15. Further
increasing the doping suppresses the magnetic order and for x = 0.2 the authors find only an onset
of spin freezing at T = 8 K.
101
800
counts per 5 min.
800
a
b
700
700
600
600
500
500
scan 1 twinned
400
0.4
0.5
0.6
400
0.7
scan 2 twinned
0.3
0.4
0.5
90
90
d
counts per 5 min.
c
80
80
70
70
60
60
50
50
40
30
0.6
40
scan 1 detwinned
30
0.4
0.5
0.6
0.7
scan 2 detwinned
0.3
0.4
0.5
0.6
r.l.u. along a*
r.l.u. along a*
FIG. 20. One dimensional nature of the magnetic fluctuations in YBa2 Cu3 O6.6 , after Mook et
al. (2000). Inelastic neutron data show a four-fold pattern of incommensurate points around the
magnetic (1/2,1/2) reciprocal lattice position. The interpretation of these measurements has been
unclear, with both striped phases and nested Fermi surfaces being possible explanations. Because
the stripe phase is expected to propagate along a single direction, only a single set of satellites
around the antiferromagnetic position should be observed. Scans of the magnetic incommensurate
scattering for the twinned and partially detwinned sample are taken at 10 K. The scan centers
of scan 1 is at (0.55, 0.55, 2) in reciprocal lattice units (rlu), while the center of scan 2 is (0.45,
0.45, 2) rlu, both along the b∗ -direction, connecting the IC peaks. Before detwinning the intensity
ratios of the satellites along the a∗ and b∗ direction are 1.05 ± 0.05. After detwinning 1.95 ± 0.21,
which is the same as the population of the domains for the two directions. The results support
one-dimensional striped excitations.
102
measured
a
calculated
0.5
ky
1.0
Γ
0.0
b
0.5
ky
1.0
-0.5
Γ
0.0
-0.5
-1.0
-1.0
-1.0
-0.5
0.0
0.5
1.0
kx
-1.0
-0.5
0.0
0.5
1.0
kx
FIG. 21. One-dimensional electronic structure in La1.28 Nd0.6 Sr0.12 CuO4 according to ARPES,
after Zhou et al. (1999). The cross defined by the drawn lines in (a) define the regions, where
the spectral weight is mainly concentrated. For comparison the conventional 2D Fermi surface
is indicated by the dashed lines. The Fermi surface expected from two perpendicular 1D stripe
domains in the 1D interpretation form a cross (b), that resembles the cross in (a) closely. Γ is the
center of the Brillouin zone.
103
ky (2π/a)
C
B
A
kx (2π/a)
FIG. 22. A schematic of the power spectrum of the Fourier transform of the function
S(E2 , E1 )(x, y, B) =
PE2
E1 [LDOS(E, x, y, B)
− LDOS(E, x, h, 0)]dE with E1 = ±1 V and
E2 = ±12 V in Bi2212, after Hoffman et al. (2002a). Peaks due to the atoms at (0, ±1) and
(±1, 0) are labelled A. Peaks due to the supermodulation are observed at B. The four peaks at
C occur only in a magnetic field and represent the vortex-induced effects at k-space locations
(0, ±1/4) and (±1/4, 0).
104
FIG. 23. The magnetic susceptibility in La5/3 Sr1/3 NiO4 , after Lee and Cheong (1997). The
effects of charge and spin order are made visible by taking the T derivative d(χT )/dT of χT (units
on the vertical scale are emu/mole). Often these effects are less pronounced. There are three
different phases denoted by A, B, and C with decreasing T . Spin order occurs in phases B and
C, but not in A. The anomalies establish that the charge-spin correlations in the three phases are
static on a time scale more than a few seconds. Charge correlations are much shorter in A than in
B or C.
105
5. The hole-doped single-layer 214-cuprates
Compared to other high temperature superconductors, the phase diagram of hole-doped
La2 CuO4 (fig. 24) gives the unit cell), or more specifically of La2−x−y Srx (Nd, Eu)y CuO4+z is
especially rich, see also the introduction. It involves the undoped antiferromagnetic phase,
the stripe phase, the d-wave superconducting and the overdoped, possibly Fermi-surface
phase. The antiferromagnetic phase has been reviewed in HMM10. Figs. 25 and 26 show
the results of the elastic neutron scattering experiment of Tranquada et al. (1995), that
spurred the interest in stripes. It appeared possible to capture the typical features of a
modulated charge and spin state in quasi-elastic neutron scattering not only in oxygen doped
non-superconducting nickelates (next section), but also in La1.48 Sr0.12 Nd0.4 CuO4 . The data
are well explained by the presence of doped stripes of holes with carry no spin and serve as
antiphase boundary for domains of antiferromagnetically aligned spins that have no charge.
Since 1997, the properties of the stripe phase have been refined by a variety of techniques, which not only include the closely related spin sensitive spectroscopy techniques of
neutron scattering, magnetic resonance and muon spin rotation, but also via charge sensitive
methods like electric transport, ARPES, and X-ray (see experimental techniques). Due to
the chemical phase separation in samples with excess oxygen, we confine our discussion to
samples with z = 0. References to neutron, NMR and other magnetic work on doped 214
before 1997 can be found in HMM10. The structural properties are briefly mentioned in
section 1 and the caption to figure 24. The theoretical background is given in section 2.
5.1. Neutron scattering
After extensive neutron research on other nickelates and cuprates, Tranquada
et al.
(1999) reexamined the degree of stripe order in La1.48 Sr0.12 Nd0.4 CuO4 and
La1.45 Sr0.15 Nd0.4 CuO4 , in which static stripe order was promoted by substituting 40% of the
La-sites by Nd. For x = 0.12 the width of the elastic magnetic peak saturates below 30 K
106
to a value that corresponds to a spin-spin correlation-length of 20 nm. Previously measured
data on charge-order peaks and these new data can be consistently explained by disorder in
stripe spacing. Above 30 K the width of the elastic peak increases, which is characteristic for
slowly fluctuating spins, see fig. 27. This phenomenon will be discussed later together with
the NMR and µSR data. Inelastic scattering measurements show that incommensurate spin
excitations survive at and above 50 K, where the elastic signal is negligible. For the x = 0.12
sample, it appears that the low temperature tetragonal (LTT) phase, which is believed to be
favorable for pinning the stripe phase, see also µSR, and the low-temperature orthorhombic
(LTO) phase coexist between 40 and 70 K. For x = 0.15 the coexistence region of both
phases is only 7 K. The sharp Bragg peaks in the LTT phase of the latter sample indicate
that the domain size exceeds 100 nm. In slightly doped La1.55 Nd0.4 Sr0.05 CuO4 Wakimoto et
al. 2001b) observed diagonal stripe correlations with the modulation direction only along
the orthorhombic b axis, just as in Nd-free La1.95 Sr0.05 CuO4 , see below.
Also in pristine La2−x Srx CuO4 signatures of static stripes have been observed. These
signatures are most developed close to x = 1/8, where the periodicity of the stripe phase
is commensurate with the lattice and might be pinned more easily. Suzuki et al. (1998)
observed magnetic superlattice peaks in neutron-diffraction measurements on orthorhombic
La1.88 Sr0.12 CuO4 , and magnetic broadening of the 136 La NMR-line below 45 K and softening
of the longitudinal sound waves along the [110] direction in the same single crystal. These
features suggest that the dynamical incommensurate spin correlation is pinned by a lattice
instability, which develops when approaching the low-temperature tetragonal phase. Kimura
et al. (1999) noted that the magnetic peak intensity first appears at the onset of superconductivity. The static magnetic correlation length exceeds 20 nm. Zn-substitution degrades
the magnetic order. Katano et al. (2000) have investigated the effects of a strong magnetic
field on the superconductivity and the static antiferromagnetic correlations. If the field is
applied perpendicular to the CuO2 plane superconductivity is severely suppressed at 10 T.
The intensity of the incommensurate magnetic peaks around the (π, π) point is increased by
as much as 50% of that at 0 T. This enhancement can be explained by the suppression of
107
the low energy spin fluctuations.
Lake et al. (2002) performed magnetic neutron diffraction to detect the spin ordering in
single crystals of underdoped La2−x Srx CuO4 with x = 0.10. The magnetic field was aligned
perpendicular to the CuO2 -planes. Like in x = 0.12, in zero field incommensurate elastic
peaks appeared in the superconducting state, which are absent above the superconducting
transition temperature of 30 K. The external field markedly enhanced the amplitude of
the signal. In the field of 14.5 T the magnetic in-plane correlation length of 40 nm is
much greater than the superconducting coherence length and the intervortex spacing, which
implies that superconductivity and antiferromagnetism coexist throughout the bulk of the
material. The finding seems to point to the existence of a magnetic quantum critical point
very close to the superconductor in the phase diagram. The neutron data combined with
resistivity measurements also strongly suggest that in a magnetic field above the critical
field Hc2 the sample becomes an incommensurate antiferromagnetic insulator.
Aeppli et al. (1997) used polarized and unpolarized neutron scattering to measure
the wave vector- and frequency-dependent magnetic fluctuations in the normal state of
La1.86 Sr0.14 CuO4 . They found nearly diverging amplitude and length scales and ω/T -scaling,
suggesting a nearby quantum critical point with a dynamical critical exponent z = 1.
The doping dependence of the low lying excitations over the hole range of dopants in
La2−x Srx CuO4 has been systematically studied by Yamada et al. (1998). A compilation
of their results with those of other groups allows to follow the doping dependence of the
incommensurability δ of the spin fluctuations, see fig. 28. The incommensurability δ is
defined as half of the peak splitting between the two peaks at (1/2, 1/2 ± δ, 0) indexed in
the tetragonal reciprocal plane (h, k, 0). The position (1/2, 1/2, 0) corresponds to an antiferromagnetic Bragg peak position in the long-range-ordered antiferromagnetic phase. It
appeared also possible to add points to the static magnetic phase diagram at very low doping
concentrations. Wakimoto et al. (1999 and 2000) performed neutron-scattering experiments
on lightly doped La2−x Srx CuO4 single crystals in both the insulating (x = 0.03, 0.04, 0.05)
and superconducting (x = 0.06) regions. Elastic magnetic peaks appeared at low tempera108
ture in all samples. Incommensurate peaks are observed only at x = 0.05, the position of
which are rotated by 450 in reciprocal space about (π, π) from those observed for x = 0.06.
Refined measurements on a single twin of x = 0.05 show that the modulation wave vector is
only along the b∗ -axis. It demonstrates that La1.95 Sr0.05 CuO4 has a one-dimensional rather
than two-dimensional static diagonal spin modulation at low temperatures. The x = 0.04
crystal has the same IC structure as the x = 0.05 crystal. δ follows the relation δ = x up
to x = 0.14. Fujita et al. (2002a) confirmed the one dimensional spin modulation along the
orthorhombic b axis for the insulating compounds with x = 0.04 and x = 0.053. Just inside
the superconducting phase x = 0.06 two pairs of incommensurate peaks are additionally
observed corresponding to the spin modulation along the tetragonal axis. In fig. 28 the
data of elastic and inelastic neutron scattering of Yamada et al. (1998), Wakimoto et al.
(1999, 2000) and Fujita et al. (2002a) are summarized. The relation between δ and hole
concentration appears linear up to 0.12, see section 2.
At high doping, not only at x = 1/8 but also at x = 1/5 static stripe ordering might
be promoted. Koike et al. (2001) report resistivity, neutron scattering and µSR data for
La2−x Srx Cu1−y Zny O4 with y=0.01. At 2 K dynamical stripe correlations of spins and holes
seem to become static.
The ordered magnetic moment µ for the same series (Wakimoto et al. 2001a) varies form
0.18 µB /Cu (x = 0.03) to 0.06 µB /Cu (x = 0.07). No significant anomaly is observed at the
insulator-superconducting boundary (x = 0.055). The elastic neutron scattering is enhanced
in the vicinity of x = 0.12, where the apparent magnetic and superconducting transitions
coincide. The ordered phases in samples with 0.03 ≤ x ≤ 0.07 have a small correlation
length ξ = 0.2 nm, while for x = 0.12 the peak is resolution limited, i.e. ξ ≥ 20 nm.
When looked at the oxygen lattice vibrations by inelastic neutron scattering, McQueeney
et al. (2001) do observe an abrupt development of new vibrations near the doping-induced
metal-insulator transition, which correlates with the electronic susceptibility measured by
photoemission. The electron-lattice coupling can be regarded as a localized one-dimensional
response of the lattice to short-ranged metallic charge fluctuations.
109
Boz̃in and coworkers (2000) resolved the atomic pair distribution function for 0 < x < 0.3
at 10 K. The in-plane Cu-O bond distribution broadens as a function of doping up to optimal
doping. Thereafter the peak abruptly sharpens. The peak can well be explained by a local
microscopic coexistence of doped and undoped material. It suggests a crossover from a
charge inhomogeneous state at and below optimal doping to a homogeneous charge state
above optimal doping. This effect correlates with the disappearance of the normal-state
pseudogap.
Also in an oxygen doped predominantly stage 4 La2 CuO4+y single crystal (onset temperature of 42 K) the low temperature magnetic scattering shows the same incommensurability
as the Sr-doped samples (Lee et al. 1999). The simultaneous appearance of the elastic
magnetic peaks and superconductivity suggest that the two phenomena are correlated. The
incommensurate wave vector appears not precisely aligned with the Cu-O-Cu tetragonal direction, but rotated by 3.30 with respect to the Cu-O-Cu direction (Y-shift), see also section
2.
Mason et al. (1996) measured the effect of superconductivity on the magnetic response
near (π, π) and at intermediate frequencies (6 < h̄ω < 15 meV). As the total moment sum
rule and the singlet nature of superconductivity suggest, they found that the suppression of
the low energy magnetic response in the superconducting state is accompanied by an increase
in the response at higher energies. For h̄ω just above the energy, where superconductivity
begins to enhance magnetic scattering, the spectral weight added by superconductivity is
extraordinary sharp, implying a new and long length scale. Lake et al. (1999) used inelastic
neutron scattering to determine the wavevector-dependence of the spin pairing. The spin gap
is wavevector independent, even though superconductivity significantly alters the wavevector
dependence of the spin fluctuations at higher energies. It shows that the spin excitations do
not parallel the charge excitations in the superconducting state, which have d-wave character
(note that the charge excitations seen in photon electron spectroscopy are not hole states
as they are seen via the emitted electrons). Lake et al. (1999) note the similarity with
Luther-Emery liquids (Luther and Emery 1974, Rokhsar and Kivelson 1988), materials with
110
gapped (triplet) spin excitations and gapless spin-zero charge excitations. According to
Morr and Pines (2000a) this effect is a direct consequence of changes in the damping of
incommensurate antiferromagnetic spin fluctuations due to the appearance of a d-wave gap
in the fermionic spectrum. In La1.875 Ba0.125−x Srx CuO4 (x = 0.05, 0.06, 0.075 and 0.085)
Fujita et al. (2002b) studied the competition of charge- and spin-density-wave order and
superconductivity. CDW and SDW order develop simultaneously and the order is stabilized
in the low temperature tetragonal phase and low-temperature less-orthorhombic phase (the
latter appears at Sr-doping levels below 0.08 and has a different symmetry (P ccn) than the
low temperature orthorhombic phase (Bmab)) and severely suppresses superconductivity.
Study of the magnetic properties around the vortex core appears to be very informative. Lake et al. (2001) found that the vortex state can be regarded as an inhomogeneous
mixture of a superconducting spin fluid and a material containing a nearly ordered antiferromagnet. Demler et al. (2001) argue this to be a sign of the proximity to a phase with
co-existing superconductivity and spin-density-wave order. Zhu et al. (2002) explain the
observations by solving self-consistently an effective hamiltonian including interactions for
both antiferromagnetic spin-density-wave and d-wave superconductivity orderings.
No signs of the magnetic resonance have been seen in doped La2−x Srx CuO4 , or more
generally in single-layered cuprates, with one notable exception. Very recently the neutron
data of He et al. (2002) showed such a resonance to be present in the single-layer compound
Tl2 Ba2 CuO6+δ with a superconducting transition temperature of 90 K. This finding restricts
the theoretical models for the origin of the resonant mode, discussed in the theoretical section
5.2. µSR and NMR/NQR
Already in early µSR and NMR experiments signs of patterned charge localization were
seen, see HMM10. For example, Borsa et al. (1995) analyzed the staggered magnetization in
La2−x Srx CuO4 , as measured by NQR and µSR for 0 < x < 0.02, invoking microsegregation
111
of holes in domain walls, separating spin-rich domains. Most of the experiments discussed
below are intended to characterize the stripe phase in more detail. For that reason samples
are selected, where already charge and spin ordering were seen by neutron scattering.
µSR. µSR data confirm the results of the neutron and X-ray studies, and in addition
allow a more detailed picture of the homogeneity of the ordered or frozen spin state. When
Nd/Eu is substituted for La in La2−x Srx CuO4 , a well developed magnetic phase develops
for x = 0.12 which involves the whole sample, while for x = 0.15 the magnetic volume
has decreased to about half, leaving the remaining volume fraction non-magnetic down to
2 K (Kojima et al. 2000). The line shape of the x = 0.12 sample is well accounted for
by a stripe model. Also in the excess oxygen sample of Lee et al. (1999), the magnetic
site fraction amounted to about 0.4 (Savici et al. 2000). Data of Nachumi et al. (1998)
on La2−x−y Srx Ndy CuO4 (x=0.125, 0.15, 0.2) and La1.875 Ba0.125−y Sry CuO4 (y = 0.025, 0.065)
show for all samples with dopant concentrations x + y ≤ 0.15 a similar static magnetic order
with 0.3 µB Cu-moment. Superconductivity and magnetic order coexist in x = 0.15. The
phase diagram of La2−x−y Srx Euy CuO4+z has been investigated by Klauss et al. (2000), see
also experimental section. In the system a low temperature tetragonal structure is present
in the entire range of doping. Following the evolution from the long range antiferromagnetic
state at x = 0 to the static magnetic stripes, they find a nonmonotonic change of the Néel
temperature with increasing x. The obtained magnetic phase diagram of the LTT phase
resembles the generic phase diagram of the cuprates where the superconductivity is replaced
by a second antiferromagnetic phase (striped antiferromagnet), see fig. 29.
Also for La2−x Srx CuO4 (and for Y1−x Cax Ba2 Cu3 O6 ), being samples without additional
Nd or Eu, a spin glass or spin/charge-ordered state has been deduced for doping values between 0.03 and 0.07 by Niedermayer et al. (1998). Even for overdoped samples with x = 0.21
Watanabe et al. (2000a) find a slowing down of the Cu-spin fluctuations below about 0.98 K
in zero field, which is a strong indication of the existence of spin/charge-ordered state.
NMR/NQR. Although the results of all mentioned techniques are very consistent in the
112
detection of striped phases, there are differences where it comes to precise values. For
example, the magnetic freezing or ordering temperature seen in neutron data is always
much higher than seen in µSR. The obvious explanation is that time scales matter. It is
especially in this respect that nuclear quadrupole data appear to be useful. As explained
in the experimental section, slowing down of the electronic spin fluctuations speeds up
the spin dephasing (and spin-lattice relaxation) rate. At magnetic phase transitions this
process normally leads to a loss of nuclear signal intensity in the experiment. However,
the corresponding nuclear magnetization or signal, when extrapolated to time zero using
the measured T2 , keeps its non-reduced value. Hunt et al. (1999) and Singer et al. (1999)
were the first to report a signal loss in connection to the formation of the stripe phase
in La2−x−y Srx (Nd, Eu)y CuO4 . Typically, the normalized signal intensity (corrected for the
apparent T2 ) started to drop around 150 K and only recovered some of its intensity below
10 K. Suh et al. (2000) and Curro et al. (2000) demonstrated that a distribution in
dephasing times could account for this effect, which was further worked out by Teitel’baum
et al. (2000). Teitel’baum et al. (2000) argued that in La2−x−y Srx Ndy CuO4 the data of T2
and wipe-out of Cu and La signals (which don’t happen at the same temperatures) could
be well explained by the same spin dynamics with some spread in spin stiffness. Fig. 30
shows the wipe-out in La and Cu NQR in La1.48 Nd0.4 Sr0.12 CuO4 . For both Cu isotopes wipe
out starts around 70 K, while for the 3 satellites of
139
La this temperature is around 40 K.
Drawn lines are fits to the equation discussed in the experimental section with numerical
constants that are consistent with the nuclear relaxation data. The value for the mean
activation energy of 143 ± 5 K is similar to the value found by Tranquada et al. (1999), see
fig. 27, of 200 ± 50 K, and points to a strongly reduced spin stiffness (an order of magnitude
smaller than in the pure antiferromagnet). This implies that the spin system should be in
the vicinity of a quantum phase-transition to a disorder state. In fig. 27 T C and TS are the
charge- and spin-ordering temperatures as revealed by neutron scattering. In µSR magnetic
freezing occurs at a much low temperature Tm ≈ 30 K . Below Tm the high temperature
description of the NQR wipe-out features (drawn lines) breaks down. In the spin frozen
113
state apparently fluctuations which much smaller activations energies play a role. Below
3 K slowing down of the Nd moments dominate the intensity loss (Teitel’baum et al. 2000).
That spin fluctuations are directly linked to the wipe-out features was also concluded by
Julien et al. (2001).
Based on similar observations, Hunt et al. (2001) constructed a phase-diagram for the
magnetic spin fluctuations as function of hole doping, see fig. 31. In this fluctuation diagram
it is also seen that down to x = 1/8, charge ordering in stripes slows down the spin fluctuations and starts the wipe-out process for the nuclear Cu spins. The data of Hunt et al.
(1999), Teitel’baum, Büchner and de Gronckel (2000), Teitel’baum et al. (2000) and Hunt
et al. (2001) also reveal that the signal recovery at low temperatures is only partial. In line
with the explanation at high temperatures, slow spin fluctuations associated with typical
features of the stripes might well be the reason for this process. Also the featureless low
temperature lineshapes (Hunt et al. 2001) are suggestive for a motional averaging process see also the next section about the nickelates, where the line shape could be fitted by static
stripes (Abu-Shiekah et al. 2001). Also Suh et al. (1998,1999) interpret the increase in
the low temperature NQR
139
La linewidth in lightly hole-doped La1.8−x Eu0.2 Srx CuO4 and
La2 Cu1−x Lix O4 in terms of local spin structures, and suggest stripes (antiphase domain
walls) in the form of mobile loops. In these experiments the structural transition from low
temperature orthorhombic to low temperature tetragonal is seen to modify the spin state.
Early NMR and NQR experiments showed the hole concentration in the CuO2 -planes of
the high-Tc superconductors to be inhomogeneous, leading to the presence of two Cu-sites
with different electrical field gradients. Haase et al. (2000) using an ingenious NMR method
found a substantial temperature dependent spatial modulation of the O and Cu parameters
of La1.85 Sr0.15 CuO4 between 10 K and 300 K with a length scale of the modulations of only
a few lattice distances. Measurements on other cuprates indicate universality of this phenomenon. Singer et al. (2002) performed
63
Cu NQR on samples enriched with
63
Cu isotope
with 0.04 ≤ x ≤ 0.16. The extent of the spatial variation ∆xhole of the local hole concentration is reflected in the relaxation rate. It is seen that the inhomogeneous distribution
114
starts below 500 - 600 K and reaches values as large as 0.5 below 150 K. The length of the
spatial variation in xhole is estimated to be larger than 3 nm. Although these features point
to phase separation, their connection to stripes is not yet clear.
5.3. Results of other magnetic techniques
In the superconducting compounds the most pronounced effect in the magnetic susceptibility, which overshadows often all other subtleties, is the diamagnetism and flux properties
of the superconducting phase. Above Tc the magnetization exhibits two-dimensional Heisenberg antiferromagnetism, e.g. Huh et al. (2001). However, by scanning SQUID microscopy
of La2−x Srx CuO4 thin films, Iguchi et al. (2001) observed inhomogeneous magnetic domains
which persist up to 80 K, well above the superconducting transition temperature of 18-19 K.
The result suggests the existence of diamagnetic regions that are precursors to the Meissner
state far above the superconducting transition temperature and can be seen as an intrinsic
tendency toward electronic inhomogeneity.
The suppression of the superconducting transition temperature in La2−x Srx CuO4 by nonmagnetic Zn ions was investigated with ESR by Finkel’stein et al. (1990). The data showed
the appearance of a new resonance line, due to the creation of magnetic moments, localized
on the Cu ions. In samples without impurities, Kochelaev et al. (1997) found a broad
but well defined single EPR line, which they attribute to a three spin polaron, consisting of two Cu2+ ions and one p hole. Kataev et al. (1998) analyzed the EPR spectra in
La2−x−y Srx Euy CuO4 and found that in the low temperature tetragonal phase, the frequency
of the spin fluctuations of the Cu ions considerably slows down in the samples with strongly
suppressed superconductivity. The effect is especially pronounced for x close to 1/8. For
x > 0.17, where superconductivity fully develops, these signatures vanish. These effects are
further discussed in section 8.
5.4. Charge sensitive techniques - X-ray, conductance and ARPES
115
Neutron scattering, muon spin rotation and nuclear quadrupolar or magnetic resonance
are obvious techniques to address magnetic properties. Because of the intimate connection
between charges and spins in striped systems (see the preceding theoretical and experimental sections), the results obtained by techniques that are especially sensitive for the charges,
like ARPES and charge transport, are relevant as well and are mentioned below.
Zhou et al.
(1999), see also Zaanen (1999a), performed ARPES experiments on
La1.28 Nd0.6 Sr0.12 CuO4 . The electronic features, see fig. 21 in the experimental section,
are consistent with other cuprates, such as a flat band at low energy near the Brillouin zone
face. However the frequency integrated spectral weight is confined inside one-dimensional
segments in the momentum space (defined by horizontal momenta |kx | = π/4 and vertical
momenta |ky | = π/4). This departure from the two dimensional Fermi surface persists to
very high energy scale. In La1.85 Sr0.15 CuO4 Zhou et al. (2001) found the same straight
segments near (π, 0) and (0, π) antinodal regions, but also identified the existence of spectral weight along the [1,1] nodal direction. This observation reveals the dual nature of the
electronic structure of stripes. On the one hand, the electrons seem to move in the strong
potentials due to the stripes. At the same time, the ARPES in the nodal directions finds an
explanation in terms of a rather ”conventional” d-wave BCS superconductor. This seemingly
paradoxical behavior still waits for a theoretical explanation. The ARPES patterns are not
only due to electronic interactions alone. From their ARPES data on a variety of high-T c
superconductors including La2−x Srx CuO4 , Lanzara et al. (2001) conclude to the ubiquitous
presence of strong coupling between electrons and phonons, see also Allen (2001).
From X-ray experiments on La2−x−y Srx (Nd, Eu)y CuO4 Zimmermann et al. (1998) found
for x = 0.12 also a sinusoidal modulation along the c∗ -axis due to long range Coulomb interaction between the stripes. In a superconducting x = 0.15 sample, Niemoller et al. (1999)
observed the disappearance of stripe order above 62 K. Data of Saini et al. (2001) reveal a
116
stepwise increase of the local lattice fluctuations below the charge order.
A rapid decrease in the magnitude of the Hall coefficient at low temperatures in
La2−x−y Ndy Srx CuO4 has been reported by Noda et al. (1999), see also Zaanen (1999a).
The presence of this effect for x < 1/8 and its absence for x > 1/8 indicate a cross-over
from one- to two-dimensional charge transport taking place at x = 1/8. An explanation
within the t − J model has been put forward by Prelovs̆ek et al. (2001). Indications of a
1D to 2D cross-over in the Hall coefficient of La1.48 Nd0.4 Sr0.12 CuO4 as function of pressure
were found by Arumugan et al. (2002). Ando et al. (2001), see fig. 32, have presented new
in-plane resistivity data on a high quality single crystals of La2−x Srx CuO4 and find that the
hole mobility at 300 K changes intriguingly similar to that of the inverse antiferromagnetic
correlation length. In a stripe picture the data show that the charge transport is influenced by the rigidity of the magnetic correlation in the magnetic domains, which probably
means that the transverse fluctuations of stripes in the antiferromagnetic environment must
be significant, consistent with the electronic liquid crystal put forward by Kivelson et al.
(1998).
Ichikawa et al. (2000) analyzed the in-plane resistivity of La2−x−y Ndy Srx CuO4 . They
found that the transition temperature for local charge ordering decreases monotonically with
x, and hence conclude that the local antiferromagnetic order is uniquely correlated with the
anomalous depression of superconductivity at x = 1/8. This result supports theories where
superconductivity depends on the existence of charge-stripe order.
5.5. Summary
Static and dynamic properties of charge and spins in La2−x−y Ndy Srx CuO4 are well documented. The magnetic resonance is not seen. Charges and spins are short range ordered
and stripes and superconductivity coexist. However, some issues remain to be solved. In fig.
31 we reproduced the compilation made by Hunt et al. (2001) and Ichikawa et al. (2000)
117
for La1.6−x Nd0.4 Srx CuO4 . The charge and magnetic order temperatures are determined by
various techniques, like NQR, X-ray and neutron diffraction studies and magnetic susceptibility and are linked to the time scale of the experiment. It is clear that below x < 0.12 the
NQR and diffraction data lead to different interpretations. These data are not necessarily
in conflict, since NQR is an inherently local probe, whereas the diffraction measurements
require substantial spatial correlations of the charge order to get detectable peaks. Also for
the origin of the spin fluctuations below 1 K, which makes the recovery of the NQR signal
only partial and removes clear antiferromagnetic signs in the line shape, a more thorough
theoretical understanding is required.
118
119
FIG. 24. The unit cell of La2−x Srx CuO4 , in the tetragonal phase a = 0.378 nm, c = 1.329 nm,
spacegroup I4/mmm. The (x, y, z) coordinates for (La,Sr) (gray), Cu (black), O1 and O2 (open
circles) are resp. (0.000, 0.000, 0.3606), (0.000, 0.000, 0.000), (0.000, 0.500, 0.000) and (0.000,
0.000, 0.1828), see HMM10.
120
k
La2-xSrxCuO4
0.5
//
NS (spin)
X-ray (charge)
//
0.5 h
FIG. 25. The modulated spin and charge wave vector in La1.6−x Nd0.4 Srx CuO4 with x = 0.12.
The magnetic peaks are characterized by the two-dimensional wave vectors (1/2 ± ², 1/2, 0) and
(1/2, 1/2 ± ², 0) with ² ≈ x, which corresponds to one hole for every 2 Cu sites. If stripes are
pinned by the atomic displacement pattern in the low temperature tetragonal (LTT) phase within
a layer, they must rotate by π/2 from one layer to another. Indeed the stripes are essential two
dimensional with modulation wave vectors (±2², 0, l) and (0, ±², 0), after Tranquada et al. (1995).
121
FIG. 26. Idealized diagram of the spin (arrows) and charge (circles) stripe pattern within a
CuO2 plane of hole-doped La2 CuO4 with nh = 1/8. Only the Cu atoms are represented - oxygen
atoms that surround the Cu atom in a square array are omitted. The arrows indicate the orientation
of the magnetic moments of the metal atoms, which are locally antiparallel. The spin direction
rotates by π (relative to a simple antiferromagnetic structure) on crossing a domain wall (antiphase
boundary). Holes are located at the anti-phase domain boundaries, which are indicated by the rows
of circles. A filled circle denotes the presence of one hole, centered on a metal site (the hole density
is assumed to be uniform along a domain wall), after Tranquada et al. (1995).
122
2
1
20
15
-2
nm
3
25
)
4
LTO L
LTT T
O
Tm (µSR)
k (10
Integrated Intensity
LTT
-1
5
TCO
0
0 10 20 30 40 50 60 70
10
5
0
0 10 20 30 40 50 60 70
T (K)
FIG. 27. The growth of the correlation length in La1.64 Nd0.4 Sr0.12 CuO4 , seen by Tranquada
et al. (1999) in elastic neutron scattering. (a) Integrated intensity of the magnetic peaks in a.u.
The resolution width is much less than the measured width. The elastic signal seen by neutron
scattering appears to survive up to approximately 50 K, while in µSR a loss of order was already
detected near 30 K. In µSR the local hyperfine field is considered to be static on a time scale of a few
µs, while neutron measurements integrate over fluctuations within a gaussian resolution window
with a full width at half maximum of approximately 0.2 THz. (b) Peak half-width κ, showing
the restricted size of the correlation length at low temperatures. Taking the inverse of κ at low
temperature gives a correlation length of 20 nm. Above 30 K κ starts to increase and the data
can be fitted to κ = κ0 + A exp[−B/T ], with κ0 = 0.00050 nm−1 , A = 0.06 nm−1 and B ∼ 200 K..
The second term on the right side has the form predicted for the 2D quantum antiferromagnetic
Heisenberg model in the renormalized regime, see text.
123
0.5
a
ktet
ktet
ktet
0.5
0.5
0.5
0.0
0.00
b
2δ
δ (r.l.u.)
0.1
2δ
0.05
xeff
ktet
0.10
FIG. 28. Sr-doping dependence of the incommensurability δ, defined as half of the peak splitting between the two peaks at (1/2, 1/2 ± δ, 0) indexed in the tetragonal plane (h, k, 0), of the
spin fluctuations seen in inelastic neutron scattering by Yamada et al. (1998) for La 2−x Srx CuO4
(open circles). Closed circles are data of Wakimoto et al. (1999, 2000) on La 2−x Srx CuO4 for
0.03 ≤ x ≤ 0.07, obtained by elastic neutron scattering. The relation between δ and hole concentration appears linear up to 0.12. At low hole concentration the modulated structure changes form
horizontal (vertical) stripes (b) to diagonal (a) as in the nickelates, see inserts. Measurements on
a single twin of x = 0.05 show that the modulation wave vector is only along the b ∗ -axis, demonstrating that these crystal have one-dimensional rather than two-dimensional static diagonal spin
modulation at low temperatures (comparable to the nickelates).
124
200
T (K)
HTT
commensurate
long range AF
LTO
LTT
100
short range static stripe
AF
AF
0
0.00
0.05
0.10
0.15
Sr content x
critcal tilt angle
300
SC
0.20
FIG. 29. Phase diagram of La1.8−x Eu0.2 Srx CuO4 , after Klauss et al. (2000). The dashed line
denotes the structural transition from low temperature orthorhombic (LTO) to low temperature
tetragonal (LTT); in the upper right corner the high temperature tetragonal (HTT) phase is just
visible. In the gray area with increasing doping values magnetic interactions lead to long range
antiferromagnetism, short range antiferromagnetism, and static stripe phases. The solid line around
x = 0.20 marks the region with diamagnetic signals due to the superconductivity. In the stripe
regions the diamagnetic signal is weak.
125
Normalized Intensity
1.0
0.8
0.6
Cu-NQR
m=7/2
m=5/2
m=3/2
0.4
0.2
0.0
Tm
0
20
40
Ts Tc
60
80
100
120
140
160
Temperature (K)
FIG. 30. Wipe-out of 139 La and 63 Cu NQR-signals in La1.48 Nd0.4 Sr0.12 CuO4 , after Teitel’baum
et al.
(2000).
The labels m = 7/2, 5/2, and 3/2 refer respectively to the (±7/2, ±5/2),
(±5/2, ±3/2), and (±3/2, ±1/2)
139 La
transitions. For both Cu isotopes wipe out starts around
70 K, while for the 3 satellites of
139 La
this temperature is around 40 K. Drawn lines are fits with
the model discussed in the experimental section with numerical constants that are found from the
relaxation data. The value for the mean activation energy of 143 ± 5 K is consistent with the value
found by Tranquada et al. (1999), see fig. 27, of 200 ± 50 K, and points to a strongly reduced
spin stiffness (an order of magnitude smaller than of the pure antiferromagnet). This implies that
the spin system should be in the vicinity of a quantum phase-transition to a disorder state. T C
and TS are the charge- and spin-ordering temperatures as revealed by neutron scattering. T m is
the magnetic ordering temperature according to µSR. Below the magnetic freezing temperature
around 30 K the high temperature description of the wipe-out feature (drawn line) breaks down.
In the spin frozen state apparently fluctuations which much smaller activations energies play a role.
Below 3 K slowing down of the Nd moments dominate the intensity loss.
126
Temperature (K)
150
2(NR)
3(µSR)
4(NR)
5(NS)
1(NR)
100
6(NMR)
7(NR)
8(NS)
9(NR)
10(NR)
11(R)
50
0
0.08
0.12
0.16
0.20
Hole doping x
FIG. 31. Phase diagram of the 214-compounds as function of hole doping. The construction is
for an important part based on wipe-out and recovery features seen in nuclear resonance (NR). The
darker gray tones indicate increasingly slow fluctuation time scales. The superconducting boundary is indicated by the lower dashed line. The other dashed line, that connects the charge-order
temperature and the Cu wipe-out inflection points is a guide to the eye. This plots is the result
of a compilation of many data points obtained on a variety of samples by various groups and
techniques, after Hunt et al. (2001) and Ichikawa et al. (2000): (1) onset of the Cu recovery in
La1.8−x Eu0.2 Srx CuO4 , (2) onset of the La recovery in La1.8−x Eu0.2 Srx CuO4 , (3) onset of the µSR
coherent precession in La1.6−x Nd0.4 Srx CuO4 , (4) onset of La wipeout in La1.8−x Eu0.2 Srx CuO4 ,
(5)long range spin order detected by neutron spectroscopy in La1.6−x Nd0.4 Srx CuO4 , (6) temperature where
139 La
1/T1 T = 0.05s−1 K−1 in La1.8−x Eu0.2 Srx CuO4 , (7) copper wipe-out inflection
point in La1.6−x Nd0.4 Srx CuO4 , (8) onset temperature for charge order in La1.6−x Nd0.4 Srx CuO4
according to neutron and X-ray scattering, (9 and 10) onset Cu wipe-out in La 1.6−x Nd0.4 Srx CuO4
and La1.8−x Eu0.2 Srx CuO4 , and (11) upturn in the (ab)-plane resistivity in La1.6−x Nd0.4 Srx CuO4 .
The NQR and diffraction data for x < 0.12 are not necessarily in conflict, since NQR is an inherently local probe, whereas the diffraction measurements require substantial spatial correlations of
the charge order to get detectable peaks.
127
2
ξAF
3
0.2
2
0.1
1
0.0
0.00
0.05
0.10
0.15
ξAF (nm)
nheρab (Vs/cm )
nheρab at 300 K
0.3
0
0.20
x
FIG. 32. x dependence of the inverse mobility 1/µ = nh eρab (open circles - left axis) as derived
from in-plane resistivity data on high quality single crystals of La2−x Srx CuO4 crystals at 300 K,
after Ando et al. (2001). The dashed line shows the x dependence of the antiferromagnetic
√
correlation length ξAF (right axis), which is reported to be 0.38/ x nm from neutron experiments
of Ino et al. (1998). The change of the hole mobility with x is similar to that of the inverse
antiferromagnetic correlation length. In a stripe picture the data show that the charge transport
is influenced by the rigidity of the magnetic correlation in the magnetic domains, which probably
means that the transverse fluctuations of stripes in the antiferromagnetic environment must be
significant, consistent with the electronic liquid crystal put forward by Kivelson et al. (1998).
128
6. Oxygen and strontium doped 214-nickelates
In the elastic neutron-scattering experiments of Tranquada et al. (1995) static stripes
were seen in La2 NiO4+δ with δ = 1/6, i.e. at hole doping of 1/3 (the unit cell is shown
in fig. 33). For that reason a comparison of the superconducting cuprates and the hardly
conducting isostructural nickelates might shed light on the role played by superconductivity
and charge localization in the formation of the striped phase. In the striped phase charge
and spin separate. In the cuprates the charged stripes have the tendency to be half filled
and the stripes run horizontally or vertically. The Cu3+ ions in the stripes carry no spin,
while the Cu2+ ions in the magnetic domains have S = 1/2. In the nickelates the charge
spin separation is different - filled charged stripes are formed of Ni3+ , which ion carries a
spin S = 1/2. The spin domains have Ni2+ -ions with S = 1. The stripes run diagonally.
The presence of the domainwall spins is an advantage for spin sensitive studies, like NMR.
Compared to the phase diagram of the oxygen doped cuprates, the oxygen doped nickelates
have much larger regions of uniform phases. While holes can be introduced by Sr or oxygen
doping, the properties of oxygen and Sr doped nickelates are not identical. Below we summarize the main results obtained in the last few years on both types of nickelates. Before
doing so, we first show details about the phase diagram under doping.
6.1. Phases and structural changes under oxygen and Sr doping
Several groups have investigated the location of the excess oxygen site, possible staging
and uniformity of the final phase in La2 NiO4+δ . In fig. 34 we reproduce the results of an
X-ray study of Rice and Buttrey (1993). Pure and mixed phases are found to alternate, but
uniform phases are clearly more frequent than in La2 CuO4 . Similar or slightly different phase
diagrams were obtained via neutron scattering by Hosoya et al. (1992) and Tranquada et al.
(1994), and via X-ray diffraction by Tamura et al. (1996). Poirot et al. (1998) traced the
role of oxygen stoichiometry in their samples (0.015 ≤ δ ≤ 0.17) by magnetic resonance. The
129
location of the excess oxygen has been refined by Mehta and Heaney (1994). In La2 NiO4.18
oxygen is located at interstitial sites equivalent to (0.183, 0.183, 0.217). Various levels of
oxygen staging were deduced from the heat capacity data of Kyômen et al. (1999) in the
temperature range 160-240 K. The time dependence of the oxygen ordering was followed in
a neutron study by Lorenzo et al. (1995).
The structural effects of Sr-substitution in La2−x Srx NiO4 reveal three distinct regimes
(Heaney et al. 1998). For 0 ≤ x ≤ 0.2 the compounds are orthorhombic. For 0.2 ≤ x ≤ 0.6
the symmetry lowers to monoclinic. For 0.6 ≤ x ≤ 1 diffraction patterns suggest phase immiscibility between a Sr-poor and a Sr-rich phase. Tang et al. (2000) compared the crystal
symmetry and the electrical properties of La2−x Ax NiO4 (A = Ca, Sr and Ba) and linked the
transfer of electron density from the dx2 to the dx2 −y2 band to the increase in the unit cell
axis a and the decrease of c.
6.2. Neutron scattering
The experimental study of stripes started by the discovery of static stripes in the oxygen doped nickelate La2 NiO4.125 by Tranquada et al. (1994). Using single-crystal neutron
diffraction, they observed not only the magnetic first and third harmonic Bragg peaks linked
to the incommensurate magnetic ordering, but also second harmonic peaks associated with
charge ordering. The magnitude of the incommensurate splitting appeared strongly T dependent. Presumably the competition between the ordering of the hole stripes and a lattice
modulation due to ordering of the interstitial oxygens leads to a devil’s staircase of ordered
phases. In subsequent studies similar stripe features were seen also at other oxygen concentrations. Hole doping of 1/4 and 1/3 (oxygen doping of 1/8 resp. 1/6) have a special
meaning due to the commensurability of the stripes with the underlying lattice (Tranquada
et al. 1995, 1997a and 1997b, Wochner et al. 1998). In Tranquada et al. (1998) charge order
has been observed up to a temperature of at least twice the value of the magnetic transition
around 110 K. The charge stripes were all oxygen centered at T > Tm , with a shift towards
130
Ni centering at T < Tm , where Tm is the magnetic ordering temperature. In the same
sample spin-wave excitations were measured by inelastic neutron scattering, Tranquada et
al. (1997a), see fig. 35. On increasing δ from 0 to 0.105, the spin-wave velocity decreases
by a factor of 5. This trend is abruptly reversed on entering the stripe-ordered phase found
at δ = 0.133. The spin-wave excitations propagating parallel to the stripes is around 60%
of that in the pristine sample, and three times greater than the velocity for δ = 0.105. In
contrast, excitations perpendicular to the stripes appear to have a greater damping. The results exhibit an energy dispersion and broadening that is qualitatively similar to that found
in the cuprate system.
Charge modulation in Sr-doped nickelates was seen in electron diffraction studies by
Chen et al. (1993). Sachan et al. (1995) reported neutron-scattering experiments that
revealed coupled incommensurate peaks arising from magnetic as well as charge ordering at
low temperatures (below 100 K) in stoichiometric La2−x Srx NiO4+δ crystals. The modulation wave vector appears to be a sensitive function of the net hole concentration p = x + 2δ.
The magnetic correlation length increases with doping and the incommensurability follows
the simple relationship ² ≈ p. Stripe ordering does not occur exclusively in commensurate
samples but also for x = 0.225, Tranquada et al. (1996). The commensurability effects
were further studied by Lee and Cheong (1997) for x = 1/3, see also the experimental
section. Upon cooling the systems undergoes three successive transitions associated with
quasi-two-dimensional commensurate charge and spin stripe ordering in the NiO 2 planes.
The two lower temperature phases are stripe lattice states with quasi-long-range in-plane
charge correlation. When the lattice of the 2D charge stripes melts, it goes through an
intermediate glass state before becoming a disordered liquid state. This glass state shows
short-range charge order without spin order, and resembles the hexactic/nematic state in
2D melting. The linear relationship between hole doping and magnetic modulation was
extended up to x = 0.5 by Yoshizawa et al. (2000). By using polarized neutron diffraction
on samples with x = 0.275 and x = 1/3, Lee et al. (2001) were able to show that the spins
in the ordered phase are canted in the NiO2 plane away from the charge and spin stripe
131
direction. In later work of Lee et al. (2002) on the x = 0.275 sample the existence of a
stripe-liquid phase could be demonstrated. The sample was selected because the spin- and
charge-ordering wave vectors do not coincide. The incommensurate magnetic fluctuations
evolve continuously through the charge-ordering temperature of 190 K. The effective damping decreases abruptly when cooling through the transition. The energy and momentum
dependence of the data can be effectively parameterized with a damped harmonic-oscillator
model describing overdamped spin waves associated with the antiferromagnetic domains defined instantaneously by the charge stripes. Signatures of strong electron-phonon coupling
were seen in the inelastic neutron data of McQueeney et al. (1999a). Tranquada et al.
(2002) noticed the similarities of the bond-stretching-phonon anomalies in La 1.69 Sr0.31 NiO4
with those in the cuprates. The absence of a collective signature makes it likely that local
interactions between charge and lattice fluctuations dominate.
6.3 µSR and NMR/NQR and magnetic susceptibility
In the oxygen doped samples Odier et al. (1999) correlated the (dc and optical) conductivity and the magnetic susceptibility on samples with well-known oxygen stoichiometries.
The susceptibility data were modelled by a Curie-Weiss law plus a cluster contribution to
account for the nearly constant T dependence between 200 and 800 K. Hasegawa et al.
(1996) found the magnetization to be anisotropic and suggest a spin glass state below 20 K.
In La2−x Srx NiO4 close to the conducting composition weakly temperature-dependent
magnetic susceptibilities and resistivities were found, similar in magnitude to those in the
isostructural metallic La2−x Srx CuO4 (Cava 1991). The magnetic data of Lee and Cheong
(1997) on La5/3 Sr1/3 NiO4 were given in the experimental section as an example, see fig.
23, and are also mentioned under neutron scattering. Ramirez et al. (1996) presented ultrasonic and specific heat measurements on La1.67 Sr0.33 NiO4 to characterize the thermodynamic transition observed at Tc = 240 K. They conclude that the possibility of independent
hole-spin behavior is consistent with the observed entropy and measured anomalies in the
132
resistivity, susceptibility and ultrasound. Also in Raman spectroscopy on La 5/3 Sr1/3 NiO4
(Blumberg et al. 1998), the formation of a superlattice and the opening of a pseudogap in
the electron-hole excitation spectra as well as two types of double spin excitations (within
the antiferromagnetic domain and across the domain wall) were observed below the charge
ordering temperature. The evolution of the two-dimensional antiferromagnetic spin correlations in the charge-ordered state even below the spin-ordering temperature was also seen
by Yamamoto et al. (1998).
µSR. Chow et al. (1996) reported µSR measurements on a series of La2−x Srx NiO4+δ
with x + 2δ > 0.4. A composition dependent magnetic transition temperature T m is found
in all samples. Below Tm clear precession signals are observed in zero applied magnetic field
indicating the existence of at least short-range magnetic order on a time scale greater than
10−8 s. Above Tm the correlation times decrease by several order of magnitude. Jestädt et
al. (1999) performed µSR on samples with Sr levels 0 < x < 1. For x = 0.33 they find
peaks in both Tm and the zero-temperature staggered magnetization. The observation is
attributed to the higher degree of localization of the holes at this doping level, see also fig.
36.
NMR/NQR. Wada et al. (1993) and Furukawa et al. (1994) followed the successive
magnetic phase transitions in La2−x Srx NiO4+δ by
internal field at the
139
139
La NQR. In the magnetic phase the
La site was found to be as large as 1.8 T. Furukawa et al. (1994)
conclude that the doped holes occupy the Ni 3dz2 orbital up to x ' 1.0. By
139
La NMR
Yoshinari et al. (1999) examined the domain formation in La5/3 Sr1/3 NiO4 . Below the chargeorder temperature their single crystal showed two magnetically distinct sites, the first located
in the domain walls and the second in the hole-free domains. Although the two regions are
spatially proximate and strongly interacting, their static and dynamic magnetic properties
are quite different. The NMR data provide specific evidence in support of the stripe-glass
hypothesis and reveals pronounced and unusual spin disorder arising from stripe defects. A
133
hall mark of glassy systems is sensitivity of the transition temperature to measurement time
scale. Yoshinari et al. (1999) find that the onset of the spin ordering (160 K) is 30 K less
than seen in neutron scattering as a consequence of this difference in time scale (µs compared
to ps for neutron scattering). At low temperatures where charge order is well developed,
the static spin order exhibits a continuous distribution of moment magnitudes and in-plane
orientations. For B ⊥ c the lineshape demonstrates that the ordered moments rotate in
response to the applied field such that the most probable spin orientation is perpendicular
to the applied field. For this particular sample, NMR seems to rule out the motion of the
domain walls below the charge order.
A δ = 1/6 sample (hole doping similar to x = 1/3) was investigated by Abu-Shiekah
et al. (1999 and 2001). Compared to the quenched potential in the Sr-doped samples the
oxygen ordered samples might be more ”clean”. Abu-Shiekah et al. found that the nickelate exhibits a fluctuation spectrum which closely parallels the fluctuations of the cuprate
stripes: although neutron-scattering experiments in La2 NiO4.13 show charge and spin freezing at 220 K resp. 110 K, the NMR data indicate that the spins in the stripes become static
only at a temperature of 2 K. These slow spin fluctuations play a much smaller role in the
analogous Sr-doped sample. As in the cuprates, the spin stiffness is strongly renormalized,
suggesting strong quantum spin fluctuations due to the proximity of a spin disordering transition, see the introduction and theoretical section. The signal recovery is only partial (even
at 0.3 K) indicating that spin dynamics is still at work at temperatures as low as 0.3 K.
Abu-Shiekah et al. (2001) demonstrated that in nickelate stripe systems the nature of the
stripe order can be deduced in detail from NMR (this in contrast to the 214-cuprates, see
Hunt et al. 2001). They find that the stripe structure is strongly solitonic, with sharply
defined charge stripes with a width which is not exceeding the lattice constant by much. Surprisingly, they look quite like the site centered stripes predicted by mean field calculations
for the nickelate system (also in La1.775 Sr0.225 NiO4 the location of the charges appeared to be
on the Ni-sites as inferred from phonon and neutron data, see Pashkevich et al. 2000a). Fig.
37 shows the spectra measured for B k ab and the simulations. Below the charge ordering
134
the line clearly has contributions from two sites (A and B). The spectra were simulated with
the same parameters as for B k c. The internal field leads to a splitting of the lines and
hence can be determined precisely. While the field on the B-site is less than 0.3 T, the field
on the A-site is about 2 T, similar to the field seen in undoped antiferromagnetically ordered
samples.
6.4 Charge sensitive techniques - X-ray, conductance and ARPES
Several studies compare the magnetic and conductance properties. Cheong et al. (1994)
found clear indications of a mutual connection in the phase transitions at hole doping of 1/3.
In the oxygen doped samples Odier et al. (1999) correlated the conductivity (dc and optical)
and the magnetic susceptibility on samples with well-known oxygen stoichiometries. The
relatively low hopping energies deduced from conductivity data in La2 NiO4.125 were ascribed
to the role of the transferred exchange interaction between the Ni2+ 3dz2 orbital and the
La3+ 6s orbital through the 2pz orbital of the apical O2− (Iguchi et al. 1999). A higher hole
doping diminishes the number of Ni2+ ions and hence decreases the exchange.
Isaacs et al.
(1994) used synchrotron X-ray scattering from La1.8 Sr0.2 NiO4 and
La1.92 Sr0.075 CuO4 to establish a direct relationship between the carriers due to Sr-doping
and strong diffuse scattering. In the nickelate the scattering is peaked at the fourfold symmetric satellite positions (±δ, ±δ, l), where the basal-plane coordinate δ varies with the outof plane coordinate l of the momentum transfer. A similar scattering pattern is observed
in the cuprate. The observations point to charged domain walls. Using high resolution
X-ray scattering the existence of quenched disordered charge stripes were established for
La5/3 Sr1/3 NiO4 . The two dimensional nature appears from the correlation lengths ξa ≈
18.5 nm, ξb ≈ 40 nm and ξc ≈ 2.5 nm and by the critical exponents of the charge stripe
transition. The charge stripe ordering did not develop long-range order even at low temperatures. Charge stripes are disordered and the length scale of the disorder is quenched.
Using near-edge X-ray diffraction, Pellegrin et al. (1996) saw evidence for the Zhang-Rice
135
character of the doped carriers and of considerable overlap between the polaron states. From
hard X-ray diffraction on La1.775 Sr0.225 NiO4 Viglinate et al. (1997) found good agreement
between the charge density modulation peaks seen in these data and in neutron data. In a
recent ARPES study (Satake et al. 2000) a large downward shift of the chemical potential
with hole doping in the high doping regime x > 0.33 is seen, while the shift is suppressed at
lower values. The suppression is attributed to segregation of doped holes. Pashkevich et al.
(2000b) conclude that the strong Fano antiresonance in the optical conductivity in striped
La1.775 Sr0.225 NiO4 is caused by Ni-O stretching motion along the stripes. The antiresonance
results from electron-phonon coupling and provides evidence for finite conductivity along
the stripes at optical frequencies.
6.5 Summary
The 214 nickelates and 214 cuprates differ greatly in conductance: below x = 1/3 the
dc conductivity of the nickelates becomes very small, while in the cuprates at low temperatures superconductivity appears. However, the two materials show a great similarity in
magnetic properties. Over a large doping regime stripes are formed of which the features
are well characterized by a variety of techniques. Like in the Sr-cuprates spin freezing in
the oxygen doped samples has a surprisingly large temperature regime of slow dynamics.
Even at 1 K the size of the stripe domains is limited to about 10 nm. The spin stiffness
is strongly renormalized due to the nearby presence of a quantum critical point. In NMR
there are differences between Sr-doped and oxygen doped samples with a similar hole count.
The quenched Sr hole-donors might be less screened than in the analogous cuprates while
oxygen diffusion in the nickelates allows the oxygen ions to form more regular structures.
At low temperatures neutron and NMR data are in favor for site centered stripes with a
strong solitonic character, at least in the 1/6 oxygen doped samples.
136
FIG. 33. The unit cell of La2−x Srx NiO4+δ , in the tetragonal phase a = 0.378 nm, c = 1.329 nm,
spacegroup I4/mmm. The (x, y, z) coordinates for (La,Sr) (gray), Ni (black), O1 and O2 (open
circles) are resp. (0.000, 0.000, 0.3606), (0.000, 0.000, 0.000), (0.000, 0.500, 0.000) and (0.000,
0.000, 0.1828), see HMM10. For δ = 0.18 the interstitial excess oxygen is located at (0.183, 0.183,
0.217) and equivalent positions (small dots) - when slowly cooled the interstitial oxygens might
form a periodic pattern. Although the structure of the nickelate is similar to La 2−x Srx CuO4 , the
occupation of the 3dz 2 orbital of the Ni ion gives a much stronger hyperfine coupling between the Ni
electron and the La nuclear spin (via the apical oxygen) than between Cu and La in La 2−x Srx CuO4 .
137
2.26
0.00
0.03
0.06
0.09
0.12
biphase region
Fmmm
2.28
F4/mmm
2.30
biphase region
2c/(a+b)
2.32
biphase region
2.34
0.15
0.18
Oxygen excess δ
FIG. 34. Evolution of the ratio of the basal plane lattice constants and the c-lattice constant
√ √
(2c/a+b) in La2 NiO4+δ (using the 2× 2×1 cell) with oxygen stoichiometry at room temperature.
Data from the low-cooled ceramic specimen with δ = 0.168 and from results of Jorgensen et al.
(1989) form the basis for the proposed high δ biphase region (after Rice and Buttrey 1993). It is
seen that with oxygen doping pure and mixed phases alternate, but uniform phases are clearly more
abundant than in La2 CuO4 . Similar, sometimes slightly different phase diagrams are reported by
other authors.
138
FIG. 35. Low-energy spin-wave spectrum dispersions measured on crystals of La 2 NiO4+δ with
δ = 0, 0.105 and 0.133. Left inset indicates directions in the (h, k, l) plane along which the dispersion
has been characterized. Main panel shows dispersions along A; results for scans along B for
δ = 0.133 are shown in the right inset. Bars indicate measured peak widths. The lines through the
data are spin-wave dispersion curves with the spin-wave velocities (h̄c) of 7 ± .8 for δ = 0.105 and
20 ± 2 for δ = 0.133. For δ = 0 the value is 34 meVnm. The two curves for each δ correspond to
in-plane and out-of-plane spin-wave modes, which have different anisotropy gaps. On increasing δ
from 0 to 0.105, the spin-wave velocity decreases by a factor of 5. This trend is abruptly reversed
on entering the stripe-ordered phase found at δ = 0.133 (after Tranquada et al. 1997a).
139
TM (K)
300
200
100
0
0.0
0.5
1.0
x
FIG. 36.
Doping dependence of the magnetic freezing or ordering temperature of
La2−x Srx NiO4+δ according to µSR data of Jestädt et al. (1999). Tm first drops with increasing x, reaches a minimum at x ∼ 0.15 and then rises again to a maximum of 205 K for x = 0.33
(different Tm values for a given x are associated with different values of δ). For larger x values
the transition temperature drops again falling to a value of 45 K at x = 1. By checking T m at the
same p = x + 2δ value, p appeared not to be the appropriate parameter for the characterization of
the order temperature. The interstitial accommodation of oxygen requires a local lattice distortion
while quite different structural consequences result from Sr substitution. Both these structural
effects are likely to have significant, but dissimilar effects on the magnetic properties.
140
B||ab, 38.2 MHz
T=107 K
a
1.0
I (a.u.)
I (a.u.)
1.0
0.5
0.0
3
0.5
4
5
0.0
3
6
4
B (T)
5
6
7
8
B (T)
1.2
1.6
Simulation B||ab,38.2 MHz
T=4.2 K
1.0
(A-site)
(B-site)
(Total)
0.8
Simulation B||ab, 38.2MHz
1.4 T=107 K
A-site
1.2
B-site
Total
1.0
I (a.u.)
I (a.u.)
b
B||ab, 38.2 MHz
T=4.2 K
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
3
4
5
0.0
3
6
B (T)
4
5
6
B (T)
7
8
FIG. 37. NMR spectra measured in La2 NiO4.17 for B k ab at high (a) and low (b) temperatures
and their simulations (after Abu Shiekah et al. 2001). Below the charge ordering the spectra for
B k c clearly have contributions from two sites (A and B). Fitting gives precise values for the
electrical field gradients but are not very sensitive for the strength of the internal field. The
spectra for B k ab were simulated with the same parameters as for B k c. The internal field
leads to a splitting of the lines and hence can be determined precisely. While the field on the
B-site is less than 0.3 T, the field on the A-site is about 2 T, similar to the field seen in undoped
antiferromagnetically ordered samples. The variation in hyperfine coupling and the values of the
internal field are strong indications for site-ordered stripes, see also the theory section.
141
7. The electron-doped single-layer compound Nd2−xCex CuO4+y
Compared to the hole-doped cuprates, less attention has been paid to electron-doped
cuprates, like Nd2−x Cex CuO4 partly because of their difficult chemistry. The unit cell differs
from La2−x Srx CuO4 in the location of the oxygen atoms outside the CuO2 plane, see fig. 38.
The antiferromagnetism in undoped Nd2 CuO4 has been described in HMM10. In systems
doped by Ce, which acts as an electron donor, the ordering temperature decreases with
increasing amount of dopant and disappears around a doping level of 0.15. The magnetism
is that of a spin-diluted Heisenberg antiferromagnet, see Vajk et al. (2002). The low
temperature ordered moment decreases almost linearly with the ordering temperature from
0.66 µB /Cu in La2 CuO4 with TN ∼ 340 K, to 0.5 µB /Cu for TN ∼ 250 K in Nd2 CuO4 and
less than 0.2 µB /Cu for Ce doped Nd2 CuO4 with transition temperatures below 150 K.
Electron- and hole-doped cuprates both have copper-oxygen planes separated by rareearth layers. Hence, from symmetry and dimensionality arguments similar superconducting
properties might be expected. However, in the transition metal oxides correlation effects
are proven to be extremely important (Zaanen et al. 1985), and these are expected to be
more dominant for electron than hole conductors, although in a more conventional band
approach (King et al 1993) these difference are less outspoken. Experimentally, asymmetry
in the electronic properties has indeed been observed. In the hole-doped samples, already
a doping level of 0.05 is sufficient to kill the magnetic state, while e.g. in Nd 2−x Cex CuO4
the magnetic state survives doping up to 0.15. Connected to this, in Nd2−x Cex CuO4 the
region where superconductivity occurs is restricted to 0.15 ≤ x ≤ 0.20 only, see fig. 39.
Still, like in the hole doped case, also in electron-doped cuprates convincing indications for
d-wave pairing have been obtained. Using a scanning superconducting quantum interference
device microscope Tsuei et al. (2000) observed the half flux quantum effect, and the dx2 −y2
anisotropy of the superconducting gap was confirmed by the ARPES data of Sato et al.
(2001), and refined in the low energy polarized electronic Raman scattering experiments
of Blumberg et al. (2002). In the overdoped state the d-wave pairing is likely changed to
142
s-wave pairing (Biswas et al. (2002), Skinta et al. (2002).
7.1. Charge sensitive data
The presence of charge-spin structures is not clear. Theoretically, arguments for the
formation of charge stripes were given in an unresticted Hartree-Fock calculation by Sadori
et al. (2000). Experimentally, a charge-order instability was suggested by Onose et al.
(1999) from the temperature variation of the optical conductivity and Raman spectra for
oxygenated antiferromagnetic and reduced superconducting crystals of Nd1.85 Ce0.15 CuO4+y .
The presence of the apical oxygens as impurities plays an important role in the realization of the superconducting or antiferromagnetic phase. In the reduced crystal the spectra
change little with T . In the spectra of the oxygenated crystal a pseudogap structure evolves
around 0.3 eV and activated infrared and Raman Cu-O phonon modes grow in intensity
below 340 K. The origin is linked to a charge-order instability induced by a minute amount
of interstitial oxygen, which seems also responsible for the absence of superconductivity, see
fig. 40. Here a strong analogy exists with the oxygen doped nickelates, where charge order
is also linked to the excess oxygens. In a later study, using crystals with various amounts
of Ce dopant, Onose et al. (2001) found that below a characteristic temperature T ∗ a notable pseudogap ∆ opens in the optical conductivity for metallic but non superconducting
samples with ∆ = 10kB T ∗ . The Drude-like component is seen to evolve concomitantly
with the pseudogap - a situation, which is reminiscent of the spin density wave gap in Cr
metal. Armitage et al. (2001) reported high-resolution photoemission spectra on superconducting Nd1.85 Ce0.15 CuO4+y . They observe regions around the Fermi surface where the
near-EF intensity is suppressed and the spectral features are broad in a manner reminiscent
of the high-energy pseudogap in the underdoped hole-doped cuprates. However, instead of
occurring near the (π, 0) region, as in the p-type materials this pseudogap falls near the intersection of the underlying Fermi surface with the antiferromagnetic Brillouin zone boundary.
In a second paper, Armitage et al. (2002), the evolution of the ARPES spectra from the
143
half filled Mott insulator to the Tc = 24 K superconductor was followed. At low doping,
the Fermi surface is an electron-pocket centered around (π, 0) with volume ∝ x. Further
doping leads to the creation of a new hole-like Fermi surface (volume ∝ (1 + x)) centered at
(π, π). Harima et al. (2001) followed the evolution of the chemical potential µ as function
of doping via core-level photoemission spectra. The shift can be deduced from these data
because the binding energy of each core level is measured relative to the chemical potential.
The result shows that µ monotonically increases with x. If the suppression of µ with doping
in La2−x Srx CuO4 is attributed to strong stripe fluctuations, the increase of µ as function of
doping in the n-type 214-compounds is consistent with the absence of stripe fluctuations.
Infrared reflectance measurements of Singley et al. (2001) signal global similarities of the
cuprate phase diagram, like the presence of the pseudogap and carrier localization.
7.2. Spin sensitive results
The magnetic order and spin correlations for a Ce-doped as grown crystal with a magnetic transition temperature of 125 K were measured by Matsuda et al. (1992), see fig.
41. The staggered moments of the Cu spins grow very gradually below 125 K, while the
correlation length saturates to about 10 nm. The T dependence above 125 K can be fitted to
renormalized classical behavior with a renormalized spin stiffness ρ = 300 meV, compared
to 800 meV in the undoped compound. Yamada et al. (1999) did not find a well-defined
incommensurate spin modulation in a sample with a Tc of 18 K. The magnetic signal appears
at (π, π) with a q-width, which is broader than in the as-grown antiferromagnetic phase. For
both systems a static magnetic order was observed in the superconducting state.
In NMR signal wipe-out effects due to slow spin dynamics occur, which are closely linked
to the formation of stripes in the 214-cuprates. Here the link to stripes is only suggestive.
Fig. 42 shows the T dependence of the signal wipe-out in Nd1.85 Ce0.15 CuO4 . The intensity is
corrected for temperature and spin-spin relaxation effects. In the superconducting (oxygen
reduced) sample the intensity, which is constant at high temperatures, starts to decrease
144
below 120 K and looses about half of the intensity at 4 K. The wipe-out of the signal is not
dependent on frequency and direction of H with respect to crystallographic axes or size of
the crystal. In the oxygenated sample the wipe-out of the Cu NMR signal starts at 200 K
and is complete at 75K. Below 25 K the signal starts to reappear again.
In hole-doped compounds the Cu wipe-out is due to a distribution in slowing-down of the
Cu-spin fluctuations rates (Suh et al. 1999, Curro et al. 2000, Hunt et al. 2000, Teitel’baum
2000, Hunt et al. 2001). In the electron-doped material wipe-out can be explained for the
same reason taking into account that Cu NMR line is much narrower in electron-doped compounds (typically by a factor 103 ), Bakharev et al. (2002). The internal field distribution
in the oxygen rich non-superconducting sample is peaked around 0 (15%) and 2 T (65%).
Such a clear difference in field distribution in that ratio is expected if the doped sites carry
no spin. The field distribution around 2 T is indicative of a disordered spin density wave.
At lower doping levels the spin structure becomes more disordered, showing patches of spins
that are antiferromagnetically coupled with a large spread in spin direction.
7.3. Summary
The magnetic properties in the electron-doped 214-cuprates are more resistant against
doping than their hole-doped analogues. Dopants seem to dilute the magnetic system in a
random way. Stripes, if present, appear hard to detect. Only around x = 0.15 a pattern
seems to develop which is reminiscent of a spin density wave. Apical oxygens seem to play an
important role in the balance between superconductivity and magnetism around x = 0.15.
145
FIG. 38. The unit cell of Nd2−x Cex CuO4 , in the tetragonal phase a = 0.378 nm, c = 1.329
nm, spacegroup I4/mmm. The (x, y, z) coordinates for (Nd,Ce) (gray), Ni (black), O1 and O2
(open circles) are resp. (0.0000, 0.0000, 0.3606), (0.0000, 0.0000, 0.0000), (0.0000, 0.5000, 0.0000)
and (0.0000, 0.0000, 0.1828), see HMM10. The location of the oxygens outside the CuO 2 plane is
different from La2−x Srx CuO4 .
146
@
>?
%'&)(*!"$#
:2
83=
24
: ;<
13:
5 7378 9
56 1324
!"$#
+-,/.
+-,/.
0
0
A'BDCDEGFH)BDEI(DEJFKLFKM3BDE
FIG. 39. Magnetic and superconducting transition-temperatures as function of doping for electron and hole-doped 214-cuprates. In the hole-doped samples already a doping level of 0.05 is
sufficient to kill the magnetic state, while e.g. in Nd2−x Cex CuO4 the magnetic state survives doping up to 0.15. Connected to this, in Nd2−x Cex CuO4 the region where superconductivity occurs is
much more restricted: only between 0.15 and 0.20.
147
Change in spectral weight
0.004
electronic gap
activated phonon
(x2)
0.002
0.000
0
200
400
T (K)
FIG. 40. Temperature variation of the change in spectral weight for Nd 1.85 Ce0.15 CuO4 (in
units of effective number of electrons) in the region of 0.145 - 0.330 eV (open circles) due to
the electronic pseudogap formation, and in the region of 0.030-0.046 eV (closed circles) due to
the activated phonon modes (Onose et al. 1999). The data at 390 K are taken as reference.
Above this temperature the conductivity data show no T dependence. The quantities vary with
temperature in an almost parallel manner (the dashed line is a guide to the eye). It is clear that
the pseudogap formation and the lattice distortion begin at about 340 K, which is much higher
than the antiferromagnetic transition (120-160 K) and the resistivity upturn (around 200 K). The
authors rule out the possibility of a conventional spin-density-wave transition for the pseudogap
formation, as the energy scale of the pseudogap (about 0.3 eV) is too high, and propose charge
ordering or its fluctuation, which is induced by a minute amount of apical oxygen as an alternative.
148
0.2
0.06
0.04
0.1
0.03
0.02
κ (r.l.u.)
MS (units of µB)
0.05
0.01
0.0
0.00
0
50
100
150
200
Temperature (K)
FIG. 41. Staggered moments of the Cu spins (drawn line - left axis) and the inverse correlation length (dashed line - right axis) for an as grown Nd1.85 Ce0.15 CuO4 crystal with a transition
temperature of 125 K (Matsuda et al. 1992). The smooth magnetization data of the raw data
show that for this sample no spin reorientation occurs (as observed in some other samples). The
magnetization intensity is decomposed according to a contribution coming from the Nd and the Cu
moments. Using data for the (101) and (103) direction the staggered Cu component is deduced.
Above 100 K the correlation length starts to decrease significantly. The dashed line corresponds
to renormalized classical behavior with a renormalized spin stiffness ρs = 300 meV, compared to
800 meV in the undoped compound.
149
FIG. 42. T dependence of the
63 Cu
signal wipe-out in Nd1.85 Ce0.15 CuO4 , after Bakharev et al.
(2002). The T dependence of the normalized integrated signal intensity is corrected for temperature
and spin-spin relaxation effects. In the superconducting oxygen-reduced (red.) sample the intensity,
which is constant at high temperatures, starts to decrease below 120 K and looses about half of
the intensity at 4 K. This effect does not originate from skin-effects. The wipe-out of the signal
is not dependent on frequency, direction of B with respect to crystallographic axes or size of the
crystal. In the oxygenated (oxy.) sample the wipe-out of the Cu-NMR signal starts at 200 K
and is complete at 75K. Below 25 K the signal reappears again. The recovered signal has a strong
resemblance with that in the nickelates. This feature, the presence of a non-magnetic and magnetic
site seen in the line profile and the distribution of the internal fields are seen as evidence for the
presence of similar stripes.
150
8. The hole-doped double-layer 123- and 124-compounds
YBa2 Cu3 O6+x (unit cell shown in fig.
43) remains nonmetallic and tetragonal for
0 ≤ x ≤ 0.4, around which value the symmetry changes to orthorhombic and superconductivity sets in. Especially for 0.4 ≤ x ≤ 0.6 the way oxygen fills the CuO-chains makes
the results dependent on the preparation procedure. The magnetism in undoped YBa2 Cu3 O6
with the CuO2 double layer is analogous to that found in the single-layered La2 CuO4 . The
spin dynamics is well described as a two-dimensional Heisenberg antiferromagnet. The Néel
temperature of the insulating phase is as high as 400 K. More precise, the compound should
be regarded as a double-layer antiferromagnet. The in-plane Cu-Cu exchange constant J
(as in HMM10 we follow the J in stead of the 2J-convention) is around 100 meV. The
ratio between J and the exchange constant between bilayers J 0 is J/J 0 ' 105 . The interaction between the planes of the bilayer Jb , which is probably of the direct exchange type, is
much stronger than J 0 : 10−2 < Jb /J < 10−1 , and also the in plane anisotropy is very weak
∆J/J ' 10−4 . Due to quantum fluctuations the ordered moment is reduced to 0.64 µB . The
magnetic excitations are well described by spin wave theory. Although the XY anisotropy
is very small, it affects the critical behavior. The staggered magnetization follows a power
law ∝ [1 − (T /TN )]β with β ' 0.25 (RossatMignod et al. 1991a, Jurgens 1990). As in the
214 compounds in slightly doped, but still magnetic materials, the coupling constants will
be renormalized.
In the 123-compounds the amount of hole doping in the CuO2 planes can be varied via the
oxygen content in the chains. For the analysis of the results the various oxygen structures
in the chains in the underdoped materials form a complication. In YBa2 Cu4 O8 this complication is avoided as double instead of a single Cu-O chains run parallel to the b axis, for
which the oxygen content is thermally stable up to 8500 C (Karpinski et al. 1988). The
orthorhombic c-axis is approximately twice as long with two rare-earth ions in the chemical
unit cell. The samples behave similarly to slightly underdoped 123-samples. In the following
they will not be addressed separately. In fig. 44 we reproduce the phase diagram of the
151
123-compounds.
For references to the original papers we refer to Johnston (1991). Here we will concentrate
on the two major issues of this contribution to the handbook: the presence and magnetism
of stripes and the so-called magnetic resonance peak.
8.1. Neutron scattering
Inelastic neutron scattering gives direct information about the spin and charge fluctuations in the material. For the 123-compounds the spectra contain several important
features, including a gap in the superconducting state, a pseudogap in the normal state, and
the magnetic resonance peak (Day et al. 1999, Fong et al. 2000. In addition the scattering
experiments have revealed a pattern of incommensurate spin fluctuations, similar to those in
the 214-compounds, consistent with the stripe picture (Mook et al. 2000). By this resolution
of the broad neutron spot at (π, π) into the incommensurate peaks the long standing puzzling asymmetry in the neutron data between the 214- and 123-compounds has disappeared.
At low doping incommensurate static charge ordering is found and the magnetic pattern is
complex with a resonance and incommensurate structure at low temperature (Mook et al.
2002). Below we start with the discussion of stripes.
Stripes. In the experimental section we reproduced the inelastic neutron-scattering data
of Mook et al. (2000), as an illustration of the one dimensional nature of the magnetic fluctuations in YBa2 Cu3 O6.6 . The data show a four-fold pattern of incommensurate points around
the magnetic (1/2,1/2) reciprocal lattice position. By comparing the results of twinned and
partially detwinned samples, the authors conclude that the stripe excitations are one dimensional. In the inelastic data of Arai et al. (1999) the spin dynamics of underdoped
YBa2 Cu3 O6.7 (Tc ≈ 67 K) revealed an incommensurate wave vector dependence with ”pillars” in the dispersion relation at the position (1/2 ± d, 1/2, 0) and (1/2, 1/2 ± d, 0), which
is the same symmetry found in the 214-compounds. The value of the incommensurability d
is about 1/8, as expected for the hole concentration in a stripe domain structure.
152
Antiferromagnetic ordering in superconducting YBa2 Cu3 O6.5 was inferred by Sidis et al.
(2001) from their polarized and unpolarized elastic neutron-scattering data. The magnetic
peak intensity exhibits a marked enhancement at the superconducting transition temperature. From µSR it appears that the staggered magnetization in this sample fluctuates on a
nanosecond time scale and seems to suggest an unusual spin density wave coexisting with
superconductivity. In their high-field NMR study of the vortices in the superconducting
state, Mitrović et al. (2001) saw strong AF fluctuations outside the core, whereas inside the
electronic states seem to have a mini-gap of about 5 meV, much sharper than the variation
of the density of states outside the vortex core. The AF features might be seen as evidence
for the presence of an underlying spin density wave or magnetic stripes even in the superconducting state (Sachdev and Zhang, 2002), see also the STM experiment of Hoffman et al.
(2002a) in La2−x Srx CuO4 . In YBa2 Cu3 O6.35 Mook, Dai and Dŏgan (2002) measured stripes
of holes lying along every eighth row of copper atoms. Using inelastic neutron scattering on
the same crystal it appeared possible to observe the incommensurate scattering even in the
region of the magnetic resonance. Only at the lowest energy (just above the gap energy of 10
meV) magnetic incommensurate scattering has the correct spacing of 1/16 reciprocal lattice
units to result from antiphase boundaries (every fourth row). Incommensurate fluctuations
and the resonance, see below, could be even observed at the same time at the resonance
energy, be it that the accuracy does not allow a quantitative description.
Resonance peak.
Already in the beginning of high-Tc inelastic neutron data on
YBa2 Cu3 O7 (Rossat-Mignod et al. 1991a, 1991b and 1992, Mook et al. 1993, Fong et
al. 1995, Bourges et al. 1996) showed the presence of a sharp magnetic collective mode,
the so-called (magnetic) resonance peak, in the superconducting state, see fig. 45. Since
then this peak has also been observed in underdoped 123-compounds (Dai et al. 1996 and
1998, Fong et al. 1997, Bourges et al. 1997), see fig. 46. As the doping decreases the peak
frequency ωr decreases, while both the peak width and its integrated intensity increase. In
the underdoped systems, a considerably broadened peak at ωr is also observed in the normal
state (Fong et al. 1997, Bourges et al. 1997). The evolution of the resonance and incommen153
surate spin fluctuations in superconducting YBa2 Cu3 O6+x as function of x has been further
studied by Arai et al (1999), see fig. 47, and Dai et al. (2001). It is confirmed that these
are general features of the spin dynamical behavior for all oxygen doping levels. Furthermore the resonance and incommensurate fluctuations appear to be intimately connected.
In some theoretical studies (see theoretical section) the resonance is interpreted as the consequence of the d-wave gap symmetry of the cuprate superconductors, while other stress
the importance of the Coulomb correlations or view the resonance as a pseudo Goldstone
mode, see theoretical section. At the low doping side the neutron-scattering experiments in
YBa2 Cu3 O6.35 Mook et al. (2002) were able to see stripes and resonance features with even
better resolution.
Impurities. The effect of small amounts of doping in the CuO2 might reveal more about
the interactions in this plane crucial for superconductivity and magnetism. Most of the
neutron scattering and NMR experiments (discussed separately below) use the S = 0 Zn 2+ or
Li+ and the S = 1 Ni2+ ions to replace the S = 1/2 Cu2+ sites. Sidis et al. (2000) compared
YBa2 (Cu0.97 Ni0.03 )3 O7 with Tc = 80 K and YBa2 (Cu0.99 Ni0.01 )3 O7 with Tc = 78 K. In the
pure system the magnetic resonance peak is at Er = 40 meV. In the Ni-substituted system,
the peak shifts to lower energy with a preserved Er /Tc ratio, while the shift is much smaller
upon Zn substitution. By contrast Zn, and not Ni, restores significant the spin fluctuations
around 40 meV in the normal state. The fact that the ratio Er /Tc for Ni-doping remains
constant, suggests that the collective mode and the superconducting gap are renormalized
in the same way, which is consistent with the spin-exciton scenario (see theoretical section).
According to Polkovnikov et al. these experiments show that Zn creates a net spin S = 1/2
at the nearest neighbor Cu ions.
Small moments. Mook et al. (2001) identified recently a new magnetic scattering feature:
small magnetic moments that increase in strength when the temperature is reduced below
T ∗ and further increase below Tc . The moments are antiferromagnetic between the Cu-O
planes with a correlation length longer than 20 nm in the ab-plane and about 4 nm along
the c-axis. These data might be explained as being due to orbital currents (Chakravarty et
154
al. 2001), see also discussion in section 3. However, care is required, as small moments have
been seen before in nuclear magnetic resonance experiments in 123-samples, where they were
shown to be due to the influence of moisture ( Dooglav et al. 1999).
8.2 µSR and NMR/NQR
From a NMR point of view YBa2 Cu3 O7−δ has been very accessible as 89 Y, 17 O and 63,65 Cu
have non-identical locations and are good nuclei (samples have to be enriched in oxygen-17
as its natural abundance is only 3.7 × 10−2 %) to perform NMR, see experimental section
for more details. Consistent data have been obtained by several groups. For references to
the early work we refer to the introductions and reviews of Mehring (1992), Asayama et
al. (1996), Berthier et al. (1996), and Brom (1998). Here we concentrate on the strongly
underdoped regime, where magnetism is most pronounced.
The low frequency (compared to the neutron data) magnetic response, as probed by
the spin-lattice relaxation rates (χ00 (ω)) and the spin-spin relaxation rates (χ0 (ω)), and
the uniform susceptibility (probed by the Knight shift) appeared to be quite unusual for a
metallic system. In normal metals the susceptibility is T independent and the relaxation rate
of the nuclei follow the Korringa relation 1/T1 T K 2 is constant (see experimental section).
The most prominent features of the underdoped superconductors are the almost featureless
decrease of the Knight shift with temperature starting at some temperature T 0 well above
Tc , see fig. 48, which behavior is the same for all nuclei (Alloul et al. (1988,1989) and
Monien et al. (1991). In contrast to the Knight shift, the T dependences of the relaxation
rates of Cu on the one hand and oxygen and yttrium on the other differ considerably, see
fig. 49 (Takigawa et al. 1991). For the Cu nuclei 1/T1 T has a maximum at (in case of
optimal doping) or above (for undoped samples) the superconducting transition (Imai et al.
1988 and 1989, Horvatic et al. 1989, Warren et al. 1989, Takigawa et al. 1991), and even
follows a Curie-Weiss-like behavior (Asayama et al. 1996). Although the Cu 2+ spins become
itinerant upon doping and the material becomes superconducting, the system remains to a
155
remarkable extent well described by the physics of localized spins. Millis, Monien and Pines
(1990) presented a phenomenological model for the explanation of the early NMR data
in YBa2 Cu3 O7−δ , which is based on the Mila-Rice-Shastry hamiltonian, presented in the
experimental section. The form factor plays a major role: oxygen and yttrium sites hardly
or do not feel the antiferromagnetic correlations, while the Cu nuclei do. The cancellation on
the oxygen sites becomes problematic if the antiferromagnetic correlation has a wave vector
that is not exactly at π, π, see experimental techniques and also Pennington et al. (2001).
The deviations from Korringa behavior of
63
T1 , already seen in the early experiments (Imai
et al. 1989), established the doping dependence of the antiferromagnetic correlations. The
strong deviations from a constant Pauli susceptibility in the Knight shift data are a sign of
a change in the relevant density of states, and hence might be interpreted as the opening
of a pseudogap. However, it is important to realize that growth of short range spin order
alone causes a reduction of χ0 (q = 0) without having any gaps (Singer and Imai 2002). By
performing
89
Y NMR and
63
Cu NQR experiments on Y1−z Caz Ba2 Cu3 O7−δ Singer and Imai
(2002) confirm that a spin gap appears as the oxygen concentration is increased from 6.0 to
6.5 (at fixed z).
The NMR phase-diagram. The decrease in the Knight shift below T0 > Tc in the underdoped compounds is related to χ(0, 0), see fig. 48. Pines (1997) thinks in terms of a
precursor to a spin density wave (indications of which might also be seen in ARPES data),
and hence favors a magnetic scenario, see fig. 50. There will be in general two distinct
groups of quasiparticles on the Fermi surface. Hot quasiparticles are located in the vicinity
of hot spots, i.e. those regions in momentum space where the magnetic Umklapp surface
intersects the Fermi surface. Because of the kinematic mismatch far away from these Umklapp surfaces, the quasi-particles do not interact with the magnetic fluctuations and behave
therefore more normally (cold spots), Williams et al. (1997) compared the Knight shift
data of many superconducting cuprates having widely different Tc,max values and different hole concentrations. The Knight shift data were analyzed using the standard expression
χs = µ B
R∞
∞
N (E)[−∂f (E)/∂E]dE, with the density of states N (E) =
156
R
δ(E −E(k))dk2 /2π 2
with the quasiparticle energies E(k) = [²(k)2 + ∆T (k)2 ]1/2 . The equation for the total gap
below Tc is supposed to be a superposition: ∆T (k)2 = ∆(k)2 + Eg (k)2 , with ∆(k) being
the superconducting and Eg (k) being the normal-state gap. A fit to the data with both
∆(k) = ∆ cos 2θ and Eg (k) = Eg cos 2θ is clearly superior to fits with an s-wave pseudo-gap.
The Eg /kB Tc,max values obtained from the NMR data are consistent with those found from
ARPES measurements (Marshall et al. 1996). The Knight shift pseudo-gap is therefore
ascribed to charge in stead of spin excitations (Williams et al. 1997, Berthier et al. 1996).
In underdoped samples the maximum in (63 T1 T )−1 , see fig. 49, at temperature T ∗ with
−1
T0 > T ∗ > Tc is followed by a simultaneous decrease of (63 T1 T )−1 and increase of 63 T2G
with
decreasing T . This suppression of the low frequency spectral weight around the antiferromagnetic wavevector is a manifestation of the opening of a pseudo gap in the spin excitations
(Pines 1997, Berthier et al. 1996). In the MMP theory outlined in the experimental section, the NMR relaxation data are sensitive probes for the magnetic correlation length ξ,
and the characteristic frequency of the spin fluctuations ωSF . The third undetermined parameter is α, which is a scale factor (units of states/eV), that relates χ Q to ξ 2 : the height
of each of the four incommensurate peaks is χQi = (α/4)ξ 2 µ2B . For YBa2 Cu3 O6+x with x
= 0, 0.63 and 1,
63
T1 T and 1/63 T2G are resp. given by [135(sK/eV2 )ωSF /α, 301(eV/s)αξ],
[145(sK/eV2 )ωSF /α, 293(eV/s)αξ] and [126(sK/eV2 )ωSF /α, 310(eV/s)αξ]. The α value for
x = 0.63 equals 8.34 and for x = 1 it is 14.8. The antiferromagnetic correlation length, as
deduced from the
63
T2G measurements, is shown in fig. 51.
Based on the NMR data Pines (1997) proposed the phasediagram of fig. 52. In the
terminology of MMP (Pines 1997), above T0 χNAFL displays mean field behavior with ωSF
and ξ −2 varying linearly with T : ωSF ∼ ξ −2 (T ) ∝ a + bT , and hence
Between T0 and T ∗ , ωSF ∼ ξ −1 (T ) ∝ c + dT , leading to
63
63
2
T1 T /T2G
is constant.
T1 T /T2G is constant. The scaling
between T0 and T ∗ is not universal (“pseudoscaling”). Below T ∗ one enters the pseudogap
phase, where ξ is frozen and ωSF increases rapidly as the temperature is decreased. At Tcr
one finds ξ(Tcr ' 2a.
157
With µSR Sonier et al. (2001) found that the presence of small magnetic fields of
electronic origin is intimately related to the pseudogap transition. For optimal doping these
weak static magnetic fields appear well below the superconducting transition temperature.
Stripes. In the 214-compounds below the charge-order temperature the stripes lead to
slow spin dynamics, and strong wipe-out features. As static charge order has not been
seen in the 123 compounds, see also the
89
Y NMR results of Bobroff et al. (2002b), it
is no surprise that such wipe-out features are absent for δ < 0.5. The wipe-out seen at
lower doping values is linked to the inhomogeneous distribution of the localized carriers.
Combined with the additional neutron data of Mook et al. (2002) this might be seen as
evidence of stripes (Kobayashi et al. 2001). Magnetic structures that possibly are linked
to spin density waves or stripes have also been seen in and around the vortex cores in 123
in the NMR experiments of Mitrović et al. (2001). Also in the experiments performed by
Haase et al. (2000) the spatial modulation of the NMR properties in 214 and 123 cuprates,
although still puzzling, are possibly connected to stripes.
By NQR on the Cu(2) nuclei (see experimental section) Krämer and Mehring (1999)
confirmed the existence of a λ-like peak in the spin-spin relaxation rate in the superconducting state of highly doped YBa2 CuO7−δ . They also observed an increasing quadrupolar
broadening of the Cu(2) NQR line below 35 K. This new feature confirmed the quadrupolar
nature of the mechanism involved in the 35 K anomaly, which is therefore connected with
some type of charge redistribution. By comparing Cu(2) and Cu(1) NQR data Grévin,
Berthier and Collin (2000) conclude that the CDW develops in the chains and that these
in-chain CDW correlations are strongly involved in the appearance of an in-plane modulated
structure below Tc
Impurity induced structures. In a series of experiments using
89
Y NMR by Alloul et al.
(1991), Mahajan et al. (1994,2000) together with SQUID susceptibility data (Mendels et al.
1999) and 7 Li resonance results (Bobroff et al. 2000) and also by
63
Cu NMR experiments
by Julien et al. (2000) the effects of impurities in the CuO2 -plane have been investigated. It
is has been shown that each Zn or Li impurity, despite having no spin, induces a local, un158
paired S = 1/2 moment on the Cu ions in its vicinity at intermediate energy scales (see also
the discussion between Bobroff et al. (2002a) and Tallon et al. (2002). A similar conclusion
was reached by Finkel’stein et al. (1990) from their EPR data in Zn doped La2−x Srx CuO4 .
In the underdoped regime this can be understood by the confining property of the host
antiferromagnet, in which the impurity is a localized ”holon” which binds the momentum of
a S = 1/2 ”spinon”. More generally, the features can be understood by taking into account
the exchange interactions between a local S = 1/2 near the Zn/Li site with the fermionic
S = 1/2 excitations of a d-wave superconductor (Polkovnikov et al. 2001).
8.3 Summary
The magnetic properties of YBa2 Cu3 O6 and heavily underdoped YBa2 Cu3 O7−δ have
many aspects in common with 214-cuprates, but there are differences. Regarding stripes,
incommensurate one-dimensional features have been seen in inelastic and elastic neutron
scattering, be it only at low doping levels. It suggests that the time scale of stripe excitations at higher doping concentrations is much shorter in the 123 than in the 214 cuprates.
The difference in time scale can also be concluded from the NMR data. In NMR some
features reminiscent of slowly fluctuating stripes have been seen only in very underdoped
materials, while the more heavily doped materials are reasonably well described in terms
of a homogenous delocalized conducting magnetic system. The presence of a small spatial structure in the NMR properties is however beyond suspicion - but its origin is still
under investigation. Another difference is the presence of the magnetic resonance in the
double layered YBa2 Cu3 O7−δ and as we will see later on, also in the bismuthates, but not
in La2−x Srx CuO4 . The various explanations are discussed in the theoretical section.
159
FIG. 43. The unit cell of YBa2−x Srx Cu3 O7−δ , in the orthorhombic phase a = 0.382 nm, b
= 0.388 nm, c = 1.168 nm, spacegroup P mmm. The (x, y, z) coordinates for Y (light gray),
(Ba,Sr) (dark gray), Cu1, Cu2 (black), O1, O2, O3 and O4 (open circles) are resp. (0.5000, 0.5000,
0.5000), (0.5000, 0.5000, 0.1843), (0.0000, 0.0000, 0.0000) and (0.0000, 0.0000, 0.3556), (0.000,
0.5000, 0.0000), (0.5000, 0.0000, 0.3779), (0.0000, 0.5000, 0.3790) and (0.0000, 0.0000, 0.1590), see
HMM10.
160
500
tetragonal
insulating
T (K)
400
300
orthorhombic
metallic
TN
Tcx4
200
100
0
YBa2Cu3O6+x
6.0
6.2
6.4
6.6
6.8
7.0
oxygen content x
FIG. 44. Phase diagram of YBa2 Cu3 O6+x , after Rossat-Mignod et al. (1991a). Five typical
regimes can be defined: (i) The pure AF-state without donated holes in the Cu(2)O 2 plane (up
to x = 0.2). (ii) The doped AF-state for 0.20 < x < 0.40. (iii) The weakly-doped metallic
state 0.40 < x < 0.50 which results in an insulator-metal transition. (iv) The strongly-doped
metallic state which develops at low temperatures two superconducting states with T c ≈ 60 K for
0.5 ≤ x ≤ 0.7 and Tc ≈ 90 K for 0.8 ≤ x ≤ 1.
161
neutron intensity
650
YBa2Cu3O6.97
Q=(0.5,0.5,5.2)
40 meV
Tc=92.5 K
600
550
500
450
400
350
nuclear contribution
background
300
250
0
50
100
150
T (K)
FIG. 45. The T dependence of the neutron intensity at the AF wave vector near the resonance
energy of h̄ω = 40 meV and Q = (0.5, 0.5, 5.2) in YBa2 Cu3 O6.97 , after Bourges et al. (1996). In
agreement with previous data by Rossat-Mignod et al. (1991a and 1991b) and Mook et al. (1993),
the resonance peak undergoes a clear change at Tc , above which temperature the scattering remains
constant within the error bars, at least up to 190 K. The line is a guide to the eye.
162
Neutron Intensity
300
250
T=4K
T = 125 K
T = 200 K
Q=(q,q,5.2)
E=35 meV
200
150
0.2
0.4
0.6
0.8
q (r.l.u.)
FIG. 46. T dependence of the magnetic resonance peak in YBa2 Cu3 O6.6 , after Bourges et al.
(1997). The major problem encountered in the inelastic neutron-scattering experiments is the
extraction of the magnetic contribution from the scattering that arises from the the nuclear lattice.
There is no unambiguous determination of the energy dependence of Imχ over a wide range of
energies. The use of polarized neutrons has even led to spurious effects like the persistence of the
resonance peak in the normal state. The major criteria to determine the magnetic signal are the
dependence of the measured intensities on the wave vector as well as the temperature. Bourges et
al. (1997) determined the dynamic spin fluctuations in YBa2 Cu3 O6.6 . The magnetic resonance,
that occurs around 40 meV in overdoped samples, is shifted to 34 meV. The ratio between T c and
Er is found to be almost independent of doping level. This finding supports the idea that the
resonance energy is proportional to the superconducting gap. The wave vector and temperature
dependencies demonstrate the magnetic origin of this signal.
163
FIG. 47. Sketch of the energy dependence of the incommensurate peaks projected along Q 2D
in YBa2 Cu3 O6.7 (with a transition temperature of 67 K), after Arai et al. (1999). The figure
is obtained by making constant energy slices at 80, 130 and 180 meV, extracted from the two
dimensional maps of the measured inelastic scattering at 20 K. The separation ∆q is about 0.16
rlu in the low-energy region, shrinks at Ec and then gradually increases above Ec . The error bars
in q (full width at half the peak height) are typically around 0.1 rlu. The resonance peak is seen
as a single peak in (Q, E) space and is centered on (1/2, 1/2, 0) at an energy E c ∼ 41 meV. The
peak might be a feature, which is not connected with the incommensurate legs, being the bottom
piece of the broad dispersive feature above Ec .
164
6
0.8
0.6
Tc
0.4
2
YBa2CuO6.63
0.2
0.0
0
100
200
-5
4
χspin (10 emu/mol Cu(2))
K(T) / [K(300)-K(0)]
1.0
0
300
T(K)
FIG. 48. The T -dependence of the normalized Knight shift of
69 Y, 63 Cu, 17 O
in underdoped
YBa2 Cu3 O6.63 with Tc = 62 K. The values for the static spin susceptibility per Cu(2) is given on
the right-hand scale. Below T0 ≈ 300 K K and χspin decrease with decreasing T . Note that the
Knight shifts have exactly the same common temperature dependence as χ0 , after Takigawa et al.
(1991) and Alloul et al. (1988 and 1989).
165
8
0.4
0.3
4
0.2
2
0.1
-1
6
-1
1/T1T (O) s K
YBa2Cu3O6.63
0
0
100
200
-1
1/T1T (Cu) s K
-1
Tc
0.0
300
T(K)
FIG. 49. Sketch of the T -dependence of (T1 T )−1 for planar Cu(2) sites (solid lines) and O(2,3)
sites (dashed lines) in underdoped YBa2 Cu3 O6.63 with Tc = 62 K, based on the data of Takigawa
et al. (1991). In the underdoped compound (63 T1 T )−1 ) has a maximum around T ∗ ≈ 150 K.
The Cu-relaxation rate is strongly influenced by magnetic correlations. The clearly different T
dependencies of the Cu and O relaxation rates are explained by the symmetry of the form factors,
that filter out the antiferromagnetic correlations at the oxygen site.
166
Brillouin
magnetic
ky
π/a
Fermi surface
0
zone
boundary
-π/a
-π/a
0
kx
π/a
FIG. 50. Model of a Fermi surface in the cuprates (solid line) and the magnetic Brillouin zone
boundary (dashed line), after Pines (1997). The intercept of the two lines marks the center of the
hot spots on the Fermi surface. These are regions near (π, 0), that can be connected by the wave
vector Qi and hence can be strongly scattered into each other. This can be seen as the precursor
to a spin density wave and favors a magnetic scenario for the explanation of the pseudogap. Cold
quasiparticles are found elsewhere , where the interaction is as in normal Fermi liquids.
167
10
ξ/a
YBa2Cu3O6.63
YBa2Cu4O8
YBa2Cu3O7
5
0
0
100
200
300
400
500
T (K)
FIG. 51. T dependence of the antiferromagnetic correlation length for YBa 2 Cu3 O6+x for
x = 0.63 and x = 1 and for YBa2 Cu4 O8 , after Barzykin and Pines (1995). The correlation
times are determined from
63 T
2G ,
the Gaussian part of the spin dephasing time of
63 Cu,
see the
discussion in the experimental section. In the 214-compounds the results obtained via NMR and
via neutron scattering compare well. For that reason the analysis is believed to give reliable results
in 123 as well.
168
T0
TN
63
T(K)
PS
63
MF
2
T1T/T2G=Cst
T1T/T2G=Cst
T
*
PG
AF
SC
x
169
xopt xover
Tc
FIG. 52. Phase diagram for the cuprate superconductors, after Pines (1997). If the NMR data
are analyzed in the context of the MMP model, the Cu-data close to (π, π) are dominated by the
antiferromagnetic correlations described by a nearly antiferromagnetic fermi-liquid susceptibility
χNAFL (see the experimental section), while far away the dynamic susceptibility is Fermi-liquid
like. The Cu spin-lattice relaxation-rate goes with T /ωSF and T2G ∝ 1/ξ. The relaxation data
show that χNAFL varies dramatically with doping and temperature through changes in χQ , ωSF
and ξ, and the dependence of ωSF on ξ. Above Tcr χNAFL displays mean field or RPA behavior
with ωSF and ξ −2 varying linearly with T . Between Tcr and a second cross over temperature T ∗ ,
χ displays z = 1 dynamic scaling behavior. The phase between Tcr and T ∗ is called pseudo-scaling
because the scaling behavior is not universal. Below T ∗ one enters the pseudogap phase, in which
the antiferromagnetic correlations become frozen, while ωSF after reaching a minimum increases
rapidly as T is further decreased. Pseudogap denotes the quasiparticle gap-like behavior found
between T ∗ and Tc , a behavior not accompanied by long-range AF order. The two cross-over
temperatures Tcr and T ∗ are also seen in the magnetic susceptibility. Detailed analysis of NMR
and INS data makes it possible to obtain a criterion for Tcr in terms of the strength of the AF
correlations ξ at Tcr : in PS ξ ≥ 2a, while in MF ξ ≤ 2a.
170
9. Other multilayered cuprates.
High-Tc cuprates with additional oxide layers of bismuth, thallium or mercury are located on the optimally or overdoped side of the phase diagram, and display no magnetic
order, see the introduction. Below we briefly give the main results for Bi-2212, one of the
best investigated structures due its stable surface properties and well defined cleaving plane.
The unconventional form of superconductivity in this high-Tc superconductor was recently
convincingly demonstrated by the optical measurements of Molegraaf et al. (2002), which
showed spectral-weight transfer from high to low energy when cooling down through the
superconducting transition temperature. This behavior contradicts that expected from traditional models, where paring is due to a lowering of the potential energy and goes together
with a small increase in the kinetic energy. Information about the thallium compounds
can be found in ”Thallium-based high-temperature superconductors” (eds. Hermann and
Jakhmi 1993). Like in the 123 and 214 systems, in Bi-2212 a magnetic resonance peak has
been observed in neutron scattering in the superconducting state.
9.1. Neutron scattering
The average structure of Bi2212 is the centrosymmetric space group Bbmb with the
unit-cell parameters a=0.541 nm, b = 0.541 nm, and c = 0.309 nm, see fig.53. The incommensurate one-dimensional modulation is given by the wave vector qs = 0.21b∗ + c∗ .
Etrillard et al. (2000) performed elastic neutron scattering on a high quality single crystal
Bi2 Sr2 CaCu2 O8+δ . The data show that the previous picture is too simple and the incommensurate structure to be made up out of two interpenetrating subsystems. In a subsequent
study (Etrillard et al. 2001) investigated the structural dynamic properties. The data indicate two acoustic-like longitudinal phonon branches along the incommensurate direction
with a weak interaction between these two subsystems.
171
9.2 µSR and NMR/NQR
After reports that stripes might have been seen in ARPES data by Shen et al. (1998) see below - Brom et al. (2000) performed NMR measurements on a low doped sample with
Tc of 60.5 K to look for possible features of slow stripe fluctuations. These NMR data do not
differ essentially from those of Ishida et al. (1998) obtained on higher doped samples and
compare well with 123-samples with higher doping levels. It means that antiferromagnetic
correlations are present, but no static or slow fluctuating spin structures develop. No wipeout effects were observed.
Watanabe et al. (2000b) applied µSR on Bi2 Sr2 Ca1−x Yx (Cu1−y Zny )2 O8+δ over a wide
range of hole concentrations. At a hole doping concentration of 1/8 there is a singularity in
the magnetic correlation between the Cu spins. The muon spin depolarization rate of the
Zn-substituted sample with hole doping of 1/8 (x = 0.3125 and y = 0.025) at 0.3 K goes
almost with B −1/2 for the magnetic field B applied along the initial muon spin direction.
From these data they conclude that the magnetically excited state of Cu spins moves in a
one-dimensional direction, which supports the evidence of stripes, see fig. 54.
9.3. Charge sensitive techniques
By using scanning tunneling microscopy (STM) Hoffman et al. (2002a) were able to determine the spatial structure of the low energy quasiparticle states associated with quantized
vortices in Bi2 Sr2 CaCu2 O8+δ , see experimental techniques. These states break continuous
translational and rotational symmetries and exhibit a checkerboard pattern with a periodicity of four unit cells and an orientation parallel to the Cu-O bonds. Their existence may
be related to the spin structure seen in the neutron-scattering experiments of Lake et al.
(2001) on the spins in the vortices of La2−x Srx CuO4 , which is explained by a pinned spin
density wave (SDW) (Demler et al. 2001) of wavelength λ. In stripes such a spin density
wave is associated with a charge modulation of the same orientation and spatial extent, but
172
which wavelength λ/2, as observed in the STM experiment, see also Sachdev and Zhang
(2002). Lang et al. (2002) report STM data on underdoped Bi2212 that reveal an apparent segregation of the electronic structure into superconducting domains that are ∼ 3 nm
in size, located in an electronically distinct background. Ni-impurity resonances are used
as markers. The resonances are present in the superconducting, but not in the pseudogap
regime (see also Zaanen 2002). Scans as function of the energy interval show that the data
are linked to the strong inhomogeneity in the samples, which leads to scattering events
that are sensitive to the peculiar gap structure. In their zero-field STM study on nearly
optimally doped Bi2212 Howald et al. (2002) show the existence of static charge-density
modulation of quasiparticle states both in the normal and superconducting state. The modulation is aligned with the Cu-O bonds, with a periodicity of four lattice constants, and
exhibits features of a two-dimensional system of line objects. According to Hoffman et al.
(2002b) quasiparticle interference, due to elastic scattering between characteristic regions
of momentum-space, provides a better explanation for the conductance modulation without appeal to another order parameter. According to the authors these scattering processes
might be a potential explanation for some other incommensurate phenomena in the cuprates
as well. STM studies have also revealed that the vortex core center has an enhanced density
of states, with a decay length of about 2 nm (Pan et al. 2000).
The STM technique has also been used to shed light on the influence magnetic (Ni) and
non-magnetic (Zn) impurities have on the high Tc superconductors, see also the 123-section.
Hudson et al. (2001), see also Flatté 2001, observe at each Ni site two d-wave impurity states
of apparently opposite spin direction, whose existence indicates that Ni retains a magnetic
moment in the superconducting state. However, the quasi-particle spectrum is predominantly non-magnetic. While the non-magnetic Zn impurity destroys superconductivity, the
superconducting energy gap and correlations are not affected by Ni. These findings are
explained by several models. Polkovnikov et al. (2001) invoke the exchange interactions
between a local S = 1/2 near the impurity site with the fermionic s = 1/2 excitations of
a d-wave superconductor. Martin et al. (2002) add that the images of resonant states are
173
the result of quantum interference of the impurity signal coming from distant paths, while
stripe pinning is seen as another important ingredient by Smith et al. (2001).
In break-junction tunneling data of Zasadzinski et al. (2001) in Bi2 Sr2 CaCu2 O8+δ (see
fig. 53) over a wide range of hole concentrations the conductance exhibits sharp dips at a
voltage Ω/e measured with respect to the superconducting gap. While phonons are collective
excitations of the lattice and therefore exist in the normal state, the collective modes relevant
in this experiment seem to develop only below Tc . The dip strength is maximal at optimal
doping and Ω scales with 4.9kTc over the entire doping range, which is close to that of the
resonance spin excitation energy seen in 123 compounds by neutron scattering by e.g. Fong
et al. (1997), Dai et al. (1999), and He et al. (2002).
ARPES data in Bi2212, like in the other high-Tc ’s, show a sudden change in the energy
dispersion curves along the (0, 0)−(π, π) direction. Lanzara et al. (2001) see this as evidence
for the ubiquitous presence of strong electron-phonon interaction. According to Johnson et
al. (2001) the features in Bi2212 are associated with a coupling to a resonant mode, see also
Eschrig and Norman (2000).
The data for Bi2 Sr2 CaCu2 O8+δ of Shen et al. (1998) show changes in the single particle
excitations that strongly depend on q when T is lowered from above to below Tc , see fig.
55. These changes extend up to an energy of about 0.3 eV or 40 kB Tc . The data suggest an
anomalous transfer of spectral weight from one momentum to another, involving a sizable
momentum transfer Q = (0, 0.45π, 0). There are several possible interpretations of the data.
The observed q dependence can be readily explained if the system has collective excitations
with real-space periodicities corresponding to Q, that are enhanced or developed below T c .
It is intriguing that Q is close to the expected momenta of charge and spin ordering in
the charge stripes that were first observed in neutron scattering from La1.46 Nd0.4 Sr0.12 CuO4 .
Taking a nominal doping of 0.18 near optimal doping and using the saturated value of
x = 1/8, q = (±0.5π, 0) or (0, ±0.5π) and q 0 = (π, π ± 0.12π) or (π ± 0.12π, π), indeed close
to the observations. Using underdoped, optimally doped and overdoped samples, Chuang et
al. (1999) find electron-like portions of the Fermi surface centered around the Γ point and
174
a depletion of spectral weight around M . The flat bands observed at other photon energies
may indicate the presence of two electronic components. This is in agreement with the data
of Feng et al. (2001) on heavily overdoped Bi2212. They found the long sought bilayer
splitting. The data favor the bilayer Hubbard model over LDA calculations. It shows that
the bilayer splitting in the superconducting state is reduced to 23% of the splitting in the
normal state. The presence of bonding and antibonding states explains also the detection
of a ”peak-dip-hump” structure in the normal state of heavily overdoped samples. Also
Kordyuk et al. (2002) stress the superposition of spectral features originating from different
electronic states to explain this feature.
9.3 Summary
Under the multilayered compounds the cleavage plane of Bi-2212 makes it very attractive
for ARPES and other surface sensitive techniques. Via ARPES it has been possible to show
the closeness of the superconducting and stripes phases. Recent STM data are an even
stronger argument for the competition between superconductivity and spin/charge order in
this compound. Till now due to the small size of the crystals the presence of a magnetic
resonance is only indirectly demonstrated.
175
FIG. 53. The tetragonal unit cell of Tl2 Ba2 CaCu2 O8 , which closely resembles that of orthorhombic Bi2 Sr2 CaCu2 O8 , with a = 0.3855 nm, and c = 29.318 nm, spacegroup I4/mmm.
The (x, y, z) coordinates for Tl (dark gray), Ba (middle gray), Ca (light gray), Cu (black), O1,
O2, O3 and O4 (open circles) are resp. (0.5000, 0.2285, 0.0522), (0.0000, 0.2537, 0.1409), (0.5000,
0.2500, 0.2500), (0.5000, 0.2498, 0.1967), (0.7500, 0.0000, 0.1950), (0.2500, 0.5000, 0.2020), (0.5000,
0.2800, 0.1220), and (0.000, 0.1500, 0.0530), see HMM10.
176
-1
depolarization rate (µs )
FIG. 54.
10
Bi2Sr2Ca1-xYx(Cu1-yZny)2O8+δ
x=0.3125, y=0.025, T=0.30 K
1
0.1
0.01
1
10
100
Longtudinal field (mT)
Longitudinal-field dependences of the muon-spin depolarization rate λ for
Bi2 Sr2 Ca1−x Yx (Cu1−y Zny )2 O8+δ with x = 0.3125 and y = 0.025 at 0.30 K (after Watanabe et
al. 2000b). The solid line is the best fit with λ ∝ B −0.63(3) , which is close to 0.5 (dotted line).
From these data the authors conclude that the magnetically excited state of Cu spins moves in a
one-dimensional direction, which gives evidence for stripes.
177
(Asc-Anorm)/Anorm (%)
2
0
-2
-4
-6
FIG.
55.
Q
0.0
0.2
Momentum-dependent
0.4
0.6
k (π/a)
spectral-weight
0.8
change
1.0
along
(0, 0)
to (π, 0) for Bi2 Sr2 CaCu2 O8+δ (after Shen et al. 1998). The quantity Asc − Anorm , where A
is the k resolved single particle spectral weight, when normalized to the normal state value A norm ,
gives the difference of the occupation probabilities in the two states [nk (s) − nk ]/nk . nk and nk (s)
represent the occupation probabilities of the normal and superconducting states, respectively. The
spectral weight is transferred from one momentum to another, with a transfer vector Q broadly
peaking between 0.4π and 0.5π. This Q value is close to the expected momenta of charge and spin
ordering in the charge stripes.
178
10. Concluding remarks
In HMM10 Johnston noticed that high-Tc cuprates and parent compounds provide a
remarkably rich variety of normal state magnetic behaviors, including short-range static
and dynamic antiferromagnetic ordering of conventional spin-glass and cluster spin-glass
types, and long-range three-dimensional order. Since 1997, when the review of HMM10 was
written, these remarks have only be proven to be correct. Especially more details have been
obtained about the so-called stripes, that have been seen in inelastic neutron scattering and
ARPES data in single and double-layered compounds and in techniques which much longer
time scales, like elastic neutron scattering, NMR/NQR, µSR, susceptibility, and transport.
The so-called Resonance Peak has an even longer experimental history than stripes.
While static stripes in the 214 cuprates were reported in 1995, the resonances in 123 were
seen already in the neutron data almost from the start (1988). Also here the experimental
development has been impressive. It has been possible to follow the resonance from optimally
doped to strongly underdoped 123, and as latest development even in a single-layer cuprate
(with Tl).
Stripes and resonance peaks form the leading theme for the discussion of the new magnetic features in the singly and multi-layered cuprates and the related nickelates presented
in this contribution. The data obtained by a variety of techniques are in good agreement
and when different results are mentioned they usually can be traced back to differences in
sample preparation e.g. leading to a difference in oxygen content or surface morphology.
In the theoretical section we have covered much of the theoretical main stream. There is a
patchwork of theoretical ideas, devised to explain some aspects of the data successfully, but
failing badly in other regards. We believe the basic understanding of the nature of the static
stripes in 214 as discussed in section 2 should be correct. Regarding the dynamical stripes
and the resonance peak the situation is more complicated. By inelastic neutron scattering
both phenomena have now been seen together in some sample as a weak resonance together
with incommensurate side branches. How these phenomena have to be explained is still
179
not settled. As mentioned in section 3, it is fair to say that the main result of 15 years of
research in high-Tc superconductivity has made it clear that the mystery is far deeper than
initially expected.
180
Acknowledgements
HBB gratefully acknowledges the contributions of Oleg Bakharev, Gregory Teitel’baum,
Evgenii Nikolaev, Oscar Bernal and especially Issa Abu-Shiekah during their stay in Leiden.
This work was supported by FOM/NWO.
181
Appendix A. Static and dynamic properties of 2D Heisenberg antiferromagnets
An overview of the theory of the Heisenberg antiferromagnets is given by Johnston in
HMM10. Here we briefly summarize the main ideas as far as relevant for those quantities
that are measured experimentally. The physical properties of 2DQHAF follow from the
nearest neighbor Heisenberg hamiltonian (the model is an idealization as it is known that
other interactions are present as well, see e.g. Katanin and Kampf (2002) for the ring exchange, and Lavrov, Komiya and Ando (2002) for the strong coupling between the magnetic
subsystem and the lattice):
Ĥ = J
X
Ŝi .Ŝj
(25)
<ij>
where < ij > refers to the summation over pairs of nearest spin neighbors. J > 0 is the
antiferromagnetic exchange coupling constant. The spin operator Ŝi is defined as:


 c↑ 
1 † †

c ↑ c ↓ σα 


2
c↓
¸
·
(26)
where σα0 s are the Pauli spin matrices:


0 1
;

σx = 

1 0


 0 −i 
;

σy = 

i 0

1 0
σz = 

0 −1




(27)
Many physical and mathematical methods have been applied to handle the above hamiltonian in order to extract quantitative parameters that match the measured quantities (Auerbach 1994). The most popular method is the non-linear sigma model (NLσM), which in
combination with spin-wave theory successfully predicts the properties of La 2 CuO4 spin
dynamics (Chakravarty et al. 1989).
The extracted parameters that can be verified experimentally include the zerotemperature staggered magnetization M (0), the Néel temperature (TN ) and the spatial
correlation length ξ(T ) in the paramagnetic state. The staggered magnetization is designated as the ordering parameter of 2DQHAF. The temperature dependence of the correla182
tion length has three distinct regimes depending on the strength of the spin-spin interaction,
which is related to the spin stiffness ρs (Chakravarty et al. 1989, Zaanen 1998).
The starting point in the analysis is to define a dimensionless coupling constant g in
order to construct a generic phase diagram for the 2DQHAF. The constant g is increased
upon disorder and doping and has the form:
h̄cΛd−1
(28)
ρs
√
where c is the spin-wave velocity and Λd−1 = 2π/a for d = 2 is a cut-off wave vector. At
g=
0 K and a critical gc , a crossover occurs between the three different regimes as shown in fig.
56 (Chakravarty et al. 1989).
The first phase to consider is the renormalized classical (RC) regime for g < gc . RC is
the only phase that has an ordered Néel ground state at T = 0 K and has been experimentally realized in many systems like the cuprates (Chakravarty et al. 1989) and nickelates
(Nakajima et al. 1995). The most important outcome of the work of Chakravarty et al.
(1989) is the prediction of the temperature dependence for the finite spin-correlation length
ξ(T ) for T > TN and g < gc :
ξ(T ) ∝ exp(
2πρs
)
T
(29)
The correlation length diverges when T → 0 K as:
ξ ≈ 0.9
h̄c
2πρs
exp(
)
kB T
T
(30)
where ρs is the actual spin stiffness renormalized by quantum fluctuations. Later Hasenfratz and Niedermayer (1991) calculated the prefactor and an extra correction term for the
exponential temperature dependence of eq. (29):
ξ(T ) =
2πρs
T
T 2
e h̄c
exp(
)[1 −
+ O(
)]
8 2πρs kB
T
2πρs
2πρs
(31)
The second regime is the quantum critical regime, which is controlled by the T ≈ 0 K
fixed critical coupling gc . Exactly at g = gc the system will behave like a three-dimensional
183
classical spin system for short length scales, before it breaks up by two-dimensional quantum
fluctuations on larger length scales. At low temperatures the correlation length equals:
ξ(T ) ≈
h̄c
kB T
(32)
The third phase with g > gc is the quantum disordered regime, which is temperature
independent as T → 0 K. In this regime the correlation length is short and a gap is opened
in the spin excitations spectrum of the order of ∆ ≈ h̄c/ξ (Chakravarty et al. 1989).
In the renormalized classical regime the constants ρs and c are not free fitting parameters,
but are calculated quantities from spin-wave theory. This implies that eq. (31) predicts the
absolute value for ξ and not only its temperature dependence. The values of c and ρs are
√
determined from ρs = Zρ (S)JS 2 and c = Zc (S)2 2JSann with J the microscopic nearest
neighboring exchange coupling and ann the nearest neighboring distance (Igarashi et al.,
1992). The functions Zρ and Zc are respectively estimated within the spin-wave theory
approximation as:
Zρ (S) = 1 −
0.235 (0.041 ± 0.03)
1
−
+ O( )3
2
2S
(2S)
2S
(33)
0.1580 0.0216
1
−
+ O( )3
2
2S
(2S)
2S
(34)
Zc (S) = 1 +
The predictions made above are in excellent agreement with experimental results extracted
from systems with S = 1/2 such as La2 CuO4 (Chakravarty et al. 1989) and Sr2 CuO2 Cl2
(Greven et al. 1994). The deviations in systems with higher spin values (S ≥ 1) will be
discussed below.
Before the work of Chakraverty et al. (1989), Haldane et al. (1983) had already conjectured that systems with integer spins behave differently than half-integer spins. Based on
topological arguments, it was shown that integer spin systems have an energy gap (nowadays
called Haldane gap) between the non-magnetic singlet state and the first magnetic triplet
state, while half-integer spin systems are gapless. According to Chakraverty et al. (1989), a
184
Haldane gap is ruled out at low energies in the g parameter region where there is a Néel ordered state at T = 0 K. At high temperatures the work of Chakraverty et al. (1989) does not
distinguish between half integer and integer spins. In the renormalized classical regime, one
would expect that such an approach will be more reliable as S is increased to higher values,
the classical limit. However, the experimental data deviate from the theoretical predictions
in systems with spin S = 1 like K2 NiF4 (Greven et al. 1995) and La2 NiO4 (Nakajima et
al. 1995), where the experimental value for ρs differs about 20% from that calculated from
spin-wave theory. The discrepancy becomes larger for systems with S = 5/2 like Rb2 MnF4
(Lee et al. 1998).
In order to understand the origin of this discrepancy, Elstner et al. (1995) calculated
the high-temperature series expansion for the Fourier transform of the spin-spin correlationz
z
function < S−q
S−q
> for all spin values in the range of 1/2 ≤ S ≤ 5/2. The results lead to
a scaling law for the correlation length given by eq. (31) as:
T /JS 2
2πZρ
eZc
T /JS 2 2
Sξ(T )
)[1
−
exp(
= √
+
O(
)]
ann
T /JS 2
4πZρ
4πZρ
16 2πZρ
(35)
When plotted in this form, the data for systems of different spin values fall on the same curve.
It was suggested by the authors that the origin for the deviation between the experimental
results and the predictions of eqs. (29) and (31), is due to a series of energy scale crossovers
observed for systems with S > 1/2. The crossovers start from renormalized classical [T <<
JS 2 , eq. (31)], to classical [JS << T << JS 2 ], and finally to Curie-Weiss [T >> JS 2 ]
regimes.
Finally, the correlation length diverges when a three-dimensional long range ordering sets
in. The long range order is due to the coupling between the adjacent planes, which is about
a factor of 10−3 − 10−5 less than the exchange coupling J between the nearest neighbors.
The staggered magnetization at T ∼ 0 K is much reduced due to the two-dimensional quantum fluctuations. As an example the maximum predicted value for the ordered magnetic
moment for the Cu spins in La2 CuO4 is about 0.6 µB (Chakravarty et al. 1989) which is
slightly larger than the experimentally measured value of 0.54 µB .
185
186
2.0
Temperature
1.5
Quantum
critical
1.0
0.5
Renormalized
classical
Quantum
disorded
0.0
0.0
0.2
0.4
Néel ordered gc
state
0.6
0.8
1.0
g
FIG. 56. Schematic phase diagram for the 2DQHAF system as a function of the coupling
strength g and temperature T , after Chakravarty, Halperin and Nelson (1998). The Néel ordered
state at T = 0 is where the long-range antiferromagnetic order sets in. The centrum quantum
critical region is controlled by the T = 0 fixed point at gc . In this region the magnetic coherence
length ξ ∝ 1/T (at low temperature) and hence diverges for T → 0. Also in the renormalized
classical regime ξ diverges as we approach zero Kelvin. For the quantum disordered region ξ
becomes independent of T for T → 0. For g > gc , the effect of finite temperature will be felt, when
one exceeds the crossover temperature Tc ≈ ∆/kB , with ∆ the energy gap in the spin excitations
spectrum. For g < gc the crossover temperature occurs when the length scale of the short range
critical fluctuations equals that of the long range antiferromagnetic magnons.
187
REFERENCES
Abrikosov, A.A., 1998, Phys. Rev. B 57, 8656.
Abu-Shiekah, I.M., O.O Bernal, A.A Menovsky, H.B Brom, and J. Zaanen, 1999, Phys. Rev.
Lett. 83, 3309.
Abu-Shiekah, I.M., O. Bakharev, H.B Brom, and J. Zaanen, 2001, Phys. Rev. Lett. 87,
237201.
Abu-Shiekah, I.M., 2001, Ph-D. thesis Leiden.
Aeppli, G., T.E. Mason, S.M. Hayden, H.A. Mook, and J. Kulda, 1997, Science 278, 1432.
Alexandrov, A.S., and P.E. Kornilovitch, 2002, J. Phys. Cond. Mat. 14, 5337.
Allen, P.B., 2001, Nature 412, 494.
Alloul, H., P. Mendels, and P. Monod, 1998, Phys. Rev. Lett. 61, 746.
Alloul, H., T. Ohno, and P. Mendels, 1989, Phys. Rev. Lett. 63, 1700.
Alloul, H., P. Mendels, H. Casalta, J.F. Marucco, and J. Arabski, 1991, Phys. Rev. Lett.
67, 3140.
Altman, E, and A. Auerbach, 2002, Phys. Rev. B 65, 104508.
Anderson, P.W., 1987, Science 235, 1196.
Ando, Y., A.N. Lavrov, S. Komiya, K. Segawa, and X.F. Sun, 2001, Phys. Rev. Lett. 87,
017001.
Anisimov, V.I., and P.H. Dederichs, 1992, Sol. Stat. Comm. 84, 241.
Arai, M., T. Nihijima, Y. Endoh, T. Egami, S. Tajima, K. Tomimoto, Y. Shiohara, M.
Takahashi, A. Garrett, and S.M. Bennington, 1999, Phys. Rev. Lett. 83, 608.
Armitage, N.P., D.H. Lu, C. Kim, A. Damascelli, K.M. Shen, F. Ronning, D.L. Feng, P.
188
Bogdanov, Z.-X. Shen, Y. Onose, Y. Taguchi, Y. Tokura, P.K. Mang, N. Kaneko, and M.
Greven, 2001, Phys. Rev. Lett. 87, 147003.
Armitage, N.P., F. Ronning, D.H. Lu, C. Kim, A. Damascelli, K.M. Shen, D.L. Feng, H.
Eisaki, Z.-X. Shen, P.K. Mang, N. Kaneko, M. Greven, Y. Onose, Y. Taguchi, and Y.
Tokura, 2002, Phys. Rev. Lett. 88, 257001.
Arovas, D. P., A. J. Berlinsky, C. Kallin, and S.C. Zhang, 1997, Phys. Rev. Lett. 79, 2871.
Arrigoni E., A.P. Harju, W. Hanke, B. Brendel, and S.A. Kivelson, 2002, Phys. Rev. B 65,
134503.
Arumugam, S., M. Môri, N. Takeshita, H. Takashima, T. Noda, H. Eisaki, and S. Uchida,
2002, Phys. Rev. Lett 88, 247001.
Asayama, K., Y. Kitaoka, G-q Zheng, and K. Ishida, 1996, Progr. NMR Spectrosc. 28, 221.
Ashcroft, N.W., and N.D. Mermin, 1976, Solid State Physics (Holt-Saunders Int., Philadelphia), chapter 2.
Auerbach, A., 1994, Interacting Electrons and Quantum Magnetism (Springer-Verlag New
York).
Bakharev, O.N., I.M. Abu-Shiekah, H.B. Brom, J. Zaanen, A.A. Nugroho, I.M. Sutjahja,
and A.A. Menovsky, 2002, to be published.
Balents, L., M.P.A. Fisher, and C. Nayak, 1998, Int. J. Mod. Phys. B 12, 1033.
Bansil, A., and M. Lindroos, 1999, Phys. Rev. Lett. 83, 5154.
Bardasis, A., and J.R. Schrieffer, 1961, Phys. Rev. 121, 1050.
Barzykin, V., and D. Pines, 1995, Phys. Rev. B 52, 13585.
Baskaran, G., Z. Zou, and P.W. Anderson, 1987, Solid State Comm. 69, 973.
Baskaran, G., and P.W. Anderson, 1998, J. Phys. Chem. Sol. 59, 1780.
189
Batista, C.D., G. Ortiz, and A.V. Balatsky, 2001, Phys. Rev. B 64, 172508.
Batlogg, B., and C.M. Varma, 2000, Phys. World 13, 33.
Bednorz, J.G., and K.A. Muller, 1986, Z. Phys. B 64, 189.
Berthier, C., M.H. Julien, M. Horvatić, and Y. Berthier, 1996, J. Phys. I (France) 6, 2205.
Bianconi, A., N.L. Saini, S. Agrestini, D. di Castro, and G. Bianconi, 2000, Int. J. Mod.
Phys. B 29-31, 3342.
Bickers, N.E., D.J. Scalapino, and S.R. White, 1989, Phys. Rev. Lett. 62, 961.
Biswas, A., P. Fournier, M.M. Qazilbash, V.N. Smolyaninova, H. Balci, and R.L. Greene,
2002, Phys. Rev. Lett. 88, 207004.
Blumberg, G., A. Koitzsch, A. Gozar, B.S. Dennis, C.A. Kendziora, P. Fournier, and R.L.
Greene, 2002, Phys. Rev. Lett. 88, 107002.
Blumberg, G., B.P. Stojković, and M.V. Klein, 1995, Phys. Rev. B 52, R15741.
Blumberg, G., M.V. Klein, and S-W. Cheong, 1998, Phys. Rev. Lett. 80, 564.
Bobroff, J., W.A. MacFarlane, H. Alloul, P. Mendels, N. Blachard, G. Collin, and J.-F.
Marucco, 1999, Phys. Rev. Lett. 83, 4381.
Bobroff, J., W.A. MacFarlane, H. Alloul, and P. Mendels, 2002a, Phys. Rev. Lett. 88, 059702.
Bobroff, J., H. Alloul, S. Ouazi, P.Mendels, A. Mahajan, N. Blanchard, G. Collin, V. Guillen,
and J.-F. Marucco, 2002b, Phys. Rev. Lett. 89, 157002.
Bogdanov, P.V., A. Lanzara, S.A. Kellar, X.J. Zhou, E.D. Lu, W.J. Zheng, G. Gu, J.-I.
Shimoyama, K. Kishio, H. Ikeda, R. Yoshizaki, Z. Hussain, and Z.X. Shen, 2000, Phys.
Rev. Lett. 85, 2581.
Borisenko, S.V., M.S. Golden, S. Legner, T. Pichler, C. Dürr, M. Knupfer, J. Fink, G. Yang,
S. Abell, and H. Berger, 2000, Phys. Rev. Lett. 84, 4453.
190
Borisenko, S.V., A.A. Kordyuk, S. Legner, C. Dürr, M. Knupfer, M.S. Golden, J. Fink, K.
Nenkov, D. Eckert, G. Yang, S. Abell, H. Berger, L. Forró, B. Liang, A. Maljuk, C.T. Lin,
and B. Keimer, 2001, Phys. Rev. B. 64, 094513.
Borsa, F., P. Carretta, J.H. Cho, F.C Chou, Q. Hu, D.C. Johnston, A. Lascialfari, D.R
Torgeson, R.J Gooding, N.M. Salem, and K.J.E. Vos, 1995, Phys. Rev. B 52, 7334.
Bosch, M., W. van Saarloos, and J. Zaanen, 2001, Phys. Rev. B 63, 092501.
Bourges, P., L.P. Regnault, Y. Sidis, and C. Vettier, 1996, Phys. Rev. B 53, 876.
Bourges, P., L.P. Regnault, Y. Sidis, J. Bossy, P. Burlet, C. Vettier, J.Y. Henry, and M.
Couach, 1997, Europhys. Lett. 38, 313.
Boz̃in, E.S., G.H. Kwei, H. Takagi, and S.J.L. Billinge, 2000, Phys. Rev. Lett. 84, 5856.
Brewer, J.H., 1995, Muon Spin Rotation/Relaxation/Resonance, in: Encyclopedia of Applied Physics 11, ed. G.L. Trigg (VCH publishers, New York) p. 23.
Brinckmann, J., and P.A. Lee, 1999, Phys. Rev. Lett. 82, 2915.
Brinckmann, J., and P.A. Lee, 2001, Phys. Rev. B 65, 014502.
Brom, H.B., 1998, in: Proc. Int. School of Physics ”Enrico Fermi” 136, eds. G. Iadonisi,
J.R. Schrieffer, and M.L. Chiofalo (IOS Press, Amsterdam) p. 77.
Brom, H.B., I.M. Abu-Shiekah, O. Bakharev, O.O. Bernal, P.H. Kes, Ming-Le, E.G. Nikolaev, G.B. Teitel’baum, A.A Nugroho, A.A. Menovsky, and J. Zaanen, 2000, Int.J. Modern
Physics B 14, 3368.
Brooks, M., 1999, New Scientist, June, 38.
Buchanan, M., 2001, Nature 409, 8.
Bulut N., and D.J. Scalapino, 1996, Phys. Rev. B 53, 5149.
Burgess, C.P., J.M. Cline, R. MacKenzie, and R. Ray, 1998, Phys. Rev. B 57, 8549.
191
Bussmann-Holder, A., A. Simon, H. Buttner, and A.R. Bishop, 2000, Phil. Mag. B 80, 1955.
Carretta, P., F. Tedoldi, A. Rigamonti, F. Galli, F. Borsa, J.H. Cho, and D.C. Johnston,
1999, Eur. Phys. J. B 10, 233.
Carlson, E.W., S.A. Kivelson, Z. Nussinov, and V.J. Emery, 1998, Phys. Rev. Lett. 57,
14704.
Carlson, E.W., V.J. Emery, S.A. Kivelson, and D. Orgad, 2002, Concepts in High Temperature Superconductivity, to appear in: The Physics of Conventional and Unconventional
Superconductors, eds. K.H. Bennemann and J.B. Ketterson (Springer-Verlag, Berlin); condmat/0206217.
Cava, R.J., B. Batlogg, T.T. Palstra, J.J. Krajewski, W.F. Peck, A.P. Ramirez, and L.W.
Rupp, 1991, Phys. Rev. B 43, 1229.
Chakravarty, S., B.I. Halperin, and D.R. Nelson, 1989, Phys. Rev. B 39, 2344.
Chakravarty, S., and H.Y. Kee, 2000, Phys. Rev. B 61, 14821.
Chakravarty, S., R.B. Laughlin, D.K. Morr, and C. Nayak, 2001, Phys. Rev. B 63, 094503.
Chen, C.H., S-W. Cheong, and A.S. Cooper, 1993, Phys. Rev. Lett. 71, 2461.
Chen, H.-D., J.-P. Hu, S. Capponi, E. Arrigino, and S.-C. Zhang, 2002, Phys. Rev. Lett. 89
137004.
Cheong, S-W., H.Y. Hwang, C.H. Chen, B. Batlogg, L.W. Rupp, and S.A. Carter, 1994,
Phys. Rev. B. 49, 7088.
Chernyshev, A.L., A.H. Castro-Neto, and A.R. Bishop, 2000, Phys. Rev. Lett. 84, 4922.
Chernyshev, A.L., S.R. White, and A.H. Castro-Neto, 2002, Phys. Rev. B 65, 214527.
Chernyshyov, O., and F. Wilczek, 2002, Phys. Rev. B 62, 4208.
Chow, K.H., P.A. Pattenden, S.J. Blundell, W. Hayes, F.L. Pratt, Th. Jestädt, M.A. Green,
192
J.E. Millburn, M.J. Rosseinsky, B. Hitti, S.R. Dunsiger, R.F. Kiefl, C. Chen, and A.J.S.
Chowdhury, 1996, Phys. Rev. B 53, R14725.
Chuang, Y.-D., A.D. Gromko, D.S. Dessau, Y. Aiura, Y. Yamaguchi, K. Oka, A.J. Arko, J.
Joyce, H. Eisaki, S.I. Uchida, K. Nakamura, and Y. Ando, 1999, Phys. Rev. Lett. 83, 3717.
Chubukov, A., 1997, Europhys. Lett. 44, 655.
Curro, N.J., P.C. Hammel, B.J. Suh, M. Hücker, B. Büchner, U. Ammerahl, and A.
Revcolevschi, 2000, Phys. Rev. Lett. 85 642.
Dagotto, E., 1994, Rev. Mod. Phys. 66, 763.
Dahm, T., Manske, D., and Tewordt, L., 1998, Phys. Rev. B 58, 12454.
Dai, P., H.A. Mook, T.B. Lindemer, and F. Doğan, 1996, Phys. Rev. Lett. 77, 5425.
Dai, P., H.A. Mook, and F. Doğan, 1998, Phys. Rev. Lett. 80, 1738.
Dai, P., H.A. Mook, S.M. Hayden, G. Aeplli, T.G. Perring, R.D. Hunt, and F. Doğan, 1999,
Science 284, 1344.
Dai, P., H.A. Mook, R.D. Hunt, and F. Doğan, 2001, Phys. Rev. B 63, 054525.
Demler, E., and S.-C. Zhang, 1995, Phys. Rev. B 55, 4126.
Demler, E., H. Kohno, and S.-C. Zhang, 1998, Phys. Rev. B 58, 5719.
Demler, E., and S.-C. Zhang, 1998, Nature 396, 733.
Demler, E., S. Sachdev, and Y. Zhang, 2001, Phys. Rev. Lett. 87, 067202.
Dessau, D.S., B. O. Wells, Z.-X. Shen, W.E. Spicer, A.J. Arko, R.S. List, D.B. Mitzi, and
A. Kapitulnik, 1992, Phys. Rev. Lett. 66, 2160.
Dimashko, Y.A., C. Morais-Smith, N. Hasselmann, and A.O. Caldeira, 1999, Phys. Rev. B
60, 88.
193
Domanski, T., and J. Ranninger, 2001, Phys. Rev. B 63, 134505.
Dombre, T., 1990, J. Phys. (Paris) 51, 847.
Dooglav, A.V., A.V. Egorov, I.R. Mukhamedshin, A.V. Savinkov, H. Alloul, J. Bobroff,
W.A. MacFarlane, P. Mendels, G. Collin, N. Blanchard, and P.G. Picard, 1999, Sov. Phys.
JETP-Letters 69, 792.
Dorneich, A., W. Hanke, E. Arrigoni, M. Troyer, and S.-C. Zhang, 2002, Phys. Rev. Lett.
88, 057003.
du Croo de Jongh, M.S.L., J.M.J. van Leeuwen, and W. van Saarloos, 2000, Phys. Rev. B
62, 14844.
Eder, R., W. Hanke, and S.C. Zhang, 1998, Phys. Rev. B 57, 13781.
Elstner, N., A. Sokol, R.R.P. Singh, M. Greven, and R. J. Birgeneau, 1995, Phys. Rev. Lett.
75, 938.
Emery, V.J., 1987, Phys. Rev. Lett. 58, 2794.
Emery, V.J. and G. Reiter, 1988, Phys. Rev. B 38, 4547 and 38, 11938.
Emery, V.J., S.A. Kivelson, and H. Q Lin, 1990, Phys. Rev. Lett. 64, 475.
Emery, V.J., and S.A. Kivelson, 1993, Physica C 209, 597.
Emery, V.J., and S.A. Kivelson, 1995, Nature 374, 434.
Emery, V.J., S. A. Kivelson, and O. Zachar, 1997, Phys. Rev. B 56, 6120.
Emery, V.J., S.A. Kivelson, and J.M. Tranquada, 1999, Proc. Natl. Acad. Sci. USA 96,
8814.
Eschrig, M, and M.R. Norman, 2000, Phys. Rev. Lett. 85, 3261.
Eskes, H., R. Grimberg, W. van Saarloos, and J. Zaanen, 1996, Phys. Rev. B 54, R724.
194
Eskes, H., O.Y. Osman, R. Grimberg, W. van Saarloos, and J. Zaanen, 1998, Phys. Rev. B
58, 6963.
Etrillard, J., P. Bourges, and C.T. Lin, 2000, Phys. Rev. B 62, 150.
Etrillard, J., P. Bourges, H.F. He, B. Keimer, B. Liang, and C.T. Lin, 2001, Europhys. Lett.
55, 201.
Feng, D.L., N.P. Armitage, D.H. Lu, A. Damascelli, J.P. Hu, P. Bogdanov, A. Lanzara, F.
Ronning, K.M. Shen, H. Eisaki, C. Kim, and Z.-X. Shen, 2001, Phys. Rev. Lett. 86, 5550.
Finkelstein, A.M., V.E. Kataev, E.F. Kukovitskii, and G.B. Teitel’baum, 1990, Physica C
168, 370.
Flatté, M.E., 2001, Nature 411, 901.
Fleck, M., A.I. Lichtenstein, E. Pavarini, and A.M. Oleś, 2000, Phys. Rev. Lett. 84, 4962.
Fleck, M., A.I. Lichtenstein, and A.M. Oleś, 2001, Phys. Rev. B. 64, 134528.
Fong, H.F., B. Keimer, P.W. Anderson, D. Reznik, F. Doğan, and I.A. Aksay, 1995, Phys.
Rev. Lett. 75, 316.
Fong, H.F., B. Keimer, D.L. Milius, and I.A. Aksay, 1997, Phys. Rev. Lett. 78, 713.
Fong, H.F., P. Bourges, Y. Sidis, L.P. Regnault, J. Bossy, A. Ivanov, D.L. Milius, I.A. Aksay,
and B. Keimer, 2000, Phys. Rev. B 61, 14773.
Fradkin, E., 1989, Field Theories of Condensed Matter Systems (Addison Wesley, Redwood
City).
Fretwell, H.M., A. Kaminski, J. Mesot, J.C. Campuzano, M.R. Norman, M. Randeria, T.
Sato, R. Gatt, T. Takahashi, and K. Kadowaki, 2000, Phys. Rev. Lett. 84, 4449.
Fujita, M., K. Yamada, H. Hiraka, P.M. Gehring, S.H. Lee, S. Wakimoto, and G. Shirane,
2002, Phys. Rev. B 65, 064505.
195
Fujita, M., H. Goka, K. Yamada, and M. Matsuda, 2002, Phys. Rev. Lett. 88, 167008.
Furukawa, Yuji, and Shinji Wada, 1994, J. Phys.: Condens. Matter 6, 8023.
Gebhard, F., 1997, The Mott Metal-Insulator Transition (Springer-Verlag Berlin).
Georges, A., and G. Kotliar, 1992, Phys. Rev. B 45, 6479.
Georges, A., G. Kotliar, W. Krauth, and M.J. Rozenberg, 1996, Rev. Mod. Phys. 68, 13.
Giamarchi, T. and C. Lhuillier, 1990, Phys. Rev. B 42, 10641.
Gora, D., K. Rosciszewski, and A.M. Oleś, 1999, Phys. Rev. B 60, 7429.
Goss Levi, B., 1998, Physics Today 51, 19.
Greiter, M., Phys. Rev. Lett. 79, 4898.
Greven, M., R.J. Birgeneau, Y. Endoh, M.A. Kastner, B. Keimer, M. Matsuda, and G.
Shirane, 1994, Phys. Rev. Lett. 72, 1096.
Greven, M., R.J. Birgeneau, Y. Endoh, M.A. Kastner, M. Matsuda, and G. Shirane, 1995,
Z. Phys. B 96, 465.
Grévin, B., Y. Berthier, and G. Collin, 2000, Phys. Rev. Lett. 85, 1310.
Haase, J., C.P. Slichter, R. Stern, C.T. Milling, and D.G. Hinks, 2000, Physica C 341-348,
1727.
Haldane, F.D.M., 1983, Phys. Rev. Lett. 50, 1153.
Harima, N., J. Matsuno, A. Fujimori, Y. Onose, Y. Taguchi, and Y. Tokura, 2001, Phys.
Rev. B 64, 220507.
Hasegawa, M., 1996, Physica B 226, 375.
Hasenfratz, P., and F. Neidermayer, 1991, Phys. Lett. B 268, 231.
Hasselmann, N., A.H. Castro Neto, C. Morais Smith, and Y. Dimashko, 1999, Phys. Rev.
196
Lett. 82, 2135.
He, H, P.Bourges, Y. Sidis, C. Ulrich, L.P. Regnault, S. Pailhès, N.S. Berzigiarova, N.N.
Kolesnikov, and B. Keimer, 2002, Science 295, 1045.
Heaney, P.J., A. Mehta, G. Sarosi, V.E. Lamberti, and A. Navrotsky, 1998, Phys. Rev. B
57, 10370.
Heeger, A.J., S. Kivelson, J.R. Schrieffer, and W.P. Su, 1988, Rev. Mod. Phys. 60, 781.
Hellberg, C.S., and E. Manousakis, 1997, Phys. Rev. Lett. 78, 4609.
Hellberg, C.S., and E. Manousakis, 1999, Phys. Rev. Lett. 83, 132.
Henley, C., 1998, Phys. Rev. Lett. 80, 3590.
Hermann, A. M., and J.V. Yakmi eds., 1993, Thallium-based High-Temperature Superconductors (Marcel Dekker, New York).
Hoffman, J.E., E.W. Hudson, K.M. Lang, V. Madhavan, S.H. Pan, H. Eisaki, S. Uchida,
and J.C. Davis, 2000a, Science 295, 466.
Hoffman, J.E., K. McElroy, D.-H. Lee, K.M. Lang, H. Eisaki, S. Uchida, and J.C. Davis,
2002b, Science 297, 1148.
Horvatić, M, P. Ségransan, C. Berthier, Y. Berthier, P. Butaud, J.Y. Henry, M. Couach, and
J.P. Chaminade, Phys. Rev. B 39, 7332.
Hosoya, S., T. Omata, K. Nakajima, K. Yamada, and Y. Endoh, 1992, Physica C 202, 188.
Hotta, T., A. Feiguin, and E. Dagotto, 2001, Phys. Rev. Lett., 86, 4922.
Howald, C., H. Eisaki, N. Kaneko, and A. Kapitulnik, 2002, cond-mat/0201546.
Hudson, E.W., K.M. Lang, V. Madhavan, S.H. Pan, H. Eisaki, S. Uchida, and J.C. Davis,
2001, Nature 411, 920.
197
Huh, Y.M., J.E. Ostenson, F. Borsa, V.G. Kogan, D.K. Finnemore, A. Vietkin, A.
Revcolevschi, and M.-H. Julien, 2001, Phys. Rev. B 63, 064512.
Hunt, A.W., P.M. Singer, K.R. Thurber, and T. Imai, 1999, Phys. Rev. Lett. 82, 4300.
Hunt, A.W., P.M. Singer, A.F. Cederström, and T. Imai, 2001, Phys. Rev. B 64, 134525.
Ichikawa, N., S. Uchida, J.M. Tranquada, T. Niemöller, P.M. Gehring, S.-H. Lee, and J.R.
Schneider, 2000, Phys. Rev. Lett. 85, 1738.
Igarashi, J.-I., 1992, Phys. Rev. B 46, 10763.
Iguchi, E., H. Satoh, H. Nakatsugawa, and F. Munakata, 1999, Physica B 270, 332.
Iguchi, I., T. Yamaguchi, and A. Sugimoto, 2001, Nature 412, 420.
Imada, M., A. Fujimori, and Y. Tokura, 1998, Rev. Mod. Phys. 70, 1059.
Imai, T., T. Shimizu, H. Yasuoka, Y. Ueda, and K. Kosuge, 1988, J. Phys. Soc. Jap. 57,
2280.
Imai, T., H. Yashuoka, T. Shimizu, Y. Ueda, K. Yoshimura, and K. Kosuge, 1989, Physica
C 162-164, 169.
Ino, A., T. Mizokawa, K. Kobayashi, A. Fujimori, T. Sasagawa, T. Kimura, K. Kishio, K.
Tamasaku, H. Eisaki, and S. Uchida, 1998, Phys. Rev. Lett. 81, 2124.
Inui, M., and P. B. Littlewood, 1991, Phys. Rev. B 44, 4415.
Isaacs, E.D., G. Aeppli, P. Zschack, S-W. Cheong, H. Williams, and D.J. Buttrey, 1994,
Phys. Rev. Lett. 72, 3421.
Ishida, K., K. Yoshida, T. Mito, Y. Tokunaga, Y. Kitaoka, K. Asayama, Y. Nakayama, J.
Shimoyama, and K. Kishio, 1998, Phys. Rev. B 58, R5960.
Jestädt, Th., K. H. Chow, S.J. Blundell, W. Hayes, F.L. Pratt, B. W. Lovett, M.A. Green,
J.E. Millburn, and M.J. Rosseinsky, 1999, Phys. Rev. B. 59, 3775.
198
Johnson, P.D., T. Valla, A.V. Fedorov, Z. Usof, B.O. Wells, Q. Li, A.R. Moodenbaugh,
G.D. Gu, N. Koshizuka, C. Kendziora, S. Jian, and D.G. Hinks, 2001, Phys. Rev. Lett. 87,
177007.
Johnston, D.C., 1991, J. Magn. Magn. Mater. 100, 218.
Johnston, D.C., 1997, in: Handbook of Magnetic Materials, Vol 10, ed. K.H.J. Buschow
(Elsevier Science, Amsterdam) p. 4.
Jorgensen, J.D., B. Dabrowski, S. Pei, and D.G. Hinks, 1989, Phys. Rev. B 40, 2187.
Julien, M.-H., T. Fehér, M. Horvatic, C. Berthier, O.N. Bakharev, P. Ségranan, G. Collin,
and J.-F. Marucco, 2001, Phys. Rev. Lett. 84, 3422.
Julien, M.-H., A. Campana, A. Rigamonti, P. Carretta, F. Borsa, P. Kuhns, A.P. Reyes,
W.G. Moulton, M. Horvatic, C. Berthier, A. Vietkin, and A. Revcolevschi, 2001, Phys.
Rev. B 63, 144508.
Jurgens, M.J.G.M., 1990, Ph-D. thesis, Leiden.
Kaminski A., S. Rosenkranz, H.M. Fretwell, J.C. Campuzano, Z. Li, H. Raffy, W.G. Cullen,
H. You, C.G. Olson, C.M. Varma, and H. Hochst, 2002, Nature 416, 610.
Kampf, A.P., D.J. Scalapino, and S.R. White, 2001, Phys. Rev. B. 64, 052509.
Kaneshita, E., M. Ichioka, and K. Machida, 2002, Phys. Rev. Lett. 88, 115501.
Karpinski, J., E. Kaldis, E. Jilek, S. Rusiecki, and B. Bucher, 1988, Nature 336, 660.
Kataev, V., B. Rameev, A. Validov, B. Büchner, M. Hückler, and R. Borowski, 1998, Phys.
Rev. B. 58, R11876.
Katanin, A.A., and A.P. Kampf, 2002, Phys. Rev. B. 666, 100403(R).
Katano, S., M. Sato, K. Yamada, T. Suzuki, and T. Fukase, 2000, Phys. Rev. B. 62, R14677.
Kato, M., K. Machida, H. Nakanishi, and M. Fujita, 1990, J. Phys. Soc. Jpn. 59, 1047.
199
Kee, H.Y., S.A. Kivelson, and G. Aeppli, 2002, Phys. Rev. Lett. 88, 257002.
Kimura, H., K. Hirota, H. Matsushita, K. Yamada, Y. Endoh, S.-H. Lee, C.F. Majkrzak, R.
Erwin, G. Shirane, M. Greven, Y.S. Lee, M.A. Kastner, and R.J. Birgeneau, 1999, Phys.
Rev. B 59, 6517.
King, D.M., Z.-X. Shen, D.S. Dessau, B.O. Wells, W.E. Spicer, A.J. Arko, D.S. Marshall, J.
DiCarlo, A.G. Loeser, C.H. Park, E.R. Ratner, J.L. Peng, Z.Y. Li, and R.L. Greene, 1993,
Phys. Rev. Lett. 70, 3159.
Kipp, L., K. Rossnagel, C. Solterbeck, T. Strasser, W. Schattke, and M. Skibowski, 1999,
Phys. Rev. Lett. 83, 5551.
Kivelson, S.A., E. Fradkin, and V.J. Emery, 1998, Nature 393, 550.
Kivelson, S.A., D.H. Lee, E. Fradkin, and V. Oganesyan, 2002, Phys. Rev. B 66, 144516.
Klauss, H.-H., W. Wagener, M. Hillberg, W. Kopmann, H. Walf, F.J. Litterst, M. Hücker,
and B. Büchner, 2000, Phys. Rev. Lett. 85, 4590.
Kobayashi, Y., T. Miyashita, M. Ambai, T. Fukamachi, and M. Sato, 2001, J. Phys. Soc.
Jap. 70, 1133.
Kochelaev, B.I., J. Sichelschmidt, B. Elschner, W. Lemor, and A. Loidl, 1997, Phys. Rev.
Lett. 79, 4274.
Koike, Y., M. Aoyama, T. Adachi, M. Akoshima, T. Kawamata, I. Watanabe, S. Ohira, W.
Higemoto, K. Nagamine, H. Kimura, K. Hirota, K. Yamada, and Y. Endoh, 2001, Physica
C 364-365, 562.
Kojima, K.M., H. Eisaki, S. Uchida, Y. Fudamoto, I.M. Gat, A. Kinkhabwala, M.I. Larkin,
G.M. Luke, and Y.J. Uemura, 2000, Physica B 289-290, 343.
Kordyuk, A.A., S.V. Borisenko, T.K. Kim, K.A. Nenkov, M. Knupfer, J. Fink, M.S. Golden,
H. Berger, and R. Follath, 2002, Phys. Rev. Lett. 89, 077003.
200
Korringa, J., 1950, Physica 16, 601.
Krämer, S., and M. Mehring, 1999, Phys. Rev. Lett. 83, 396.
Kusmartsev, F.V., D. Di Castro, G. Bianconi, and A. Bianconi, 2000, Phys. Lett. A 275,
118.
Kusmartsev, F.V., 2000, Phys. Rev. Lett. 84, 530.
Kusmartsev, F.V., 2001, Europhys. Lett. 54, 786.
Kyomen, T., M. Oguni, M. Itoh, and K. Kitayama, 1999, Phys. Rev. B 60, 14841.
Lake, B., G. Aeppli, T.E. Mason, A. Schröder, D.F. McMorrow, K. Lefmann, M. Isshiki, M.
Nohara, H. Takagi, and S.M. Hayden, 1999, Nature 400, 43.
Lake, B., G. Aeppli, K.N. Clausen, D.F. McMorrow, K. Lefmann, N.E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T.E. Mason, and A. Schröder, 2001, Science 291, 1759.
Lake, B., H.M. Ronnow, N.B. Christensen, G. Aeppli, K. Lefmann, D.F. McMorrow,P.
Vorderwisch, P. Smelbidl, N. Mangkorntong, T. Sasagawa, M. Nohara, H. Takagi, and T.E.
Mason, 2002, Nature 415, 299.
Lang, K.M., V. Madhavan, J.E. Hoffman, E.E. Hudson, H. Elsaki, S. Uchida, and J.C. Davis,
2002, Nature 415, 412.
Lanzara, A., P.V. Bogdanov, X.J. Zhou, S.A. Kellar, D.F. Feng, E.D. Lu, T. Yoshida, H.
Eisaki, A. Fujiimori, H. Kishio, J.-I. Shimoyama, T. Noda, S. Uchida, Z. Hussain, and Z.X.
Shen, 2001, Nature 412, 510.
Laughlin, R.B., and D. Pines, 2000, Proc. Natl. Acad. Sci. 97, 28.
Laughlin, R.B., D. Pines, J. Schmalian, B.P. Stojkovic, and P. Woynes, 2000, Proc. Natl.
Acad. Sci. 97, 32.
Lavagna, M., and G. Stemmann, 1994, Phys. Rev. B 49, 4235.
201
Lavrov, A.N., S. Komiya, and Y. Ando, 2002, Nature 418, 385.
Lee, S.-H., and S-W. Cheong, 1997, Phys. Rev. Lett. 79, 2514.
Lee, Y.S., M. Greven, B.O. Wells, R.J. Birgeneau, and G. Shirane, 1998, Eur. Phys. J. B
96, 15.
Lee, Y.S., R.J. Birgenau, M.A. Kastner, Y. Endoh, S. Wakimoto, K. Yamada, E.W. Erwin,
S.-H. Lee, and G. Shirane, 1999, Phys. Rev. B 60, 3643.
Lee, S.-H., S-W. Cheong, K. Yamada, and C.F. Majkrzak, 2001, Phys. Rev. B 63, 060405(R).
Lee, S.-H., J.M. Tranquada, K. Yamada, D.J. Buttrey, Q. Li, and S-W. Cheong, 2002, Phys.
Rev. Lett. 88, 126401.
Lercher, M.J., and J.M. Wheatley, 1994, Phys. Rev. B 49, 736.
Lindroos, M., S. Sahrakorpi, and A. Bansil, 2002, Phys. Rev. B 65, 054514.
Littlewood, P.B., J. Zaanen, G. Aeppli, and H. Monien, 1993, Phys. Rev. B 48, 487.
Liu, D.Z., Y. Zha, and K. Levin, 1995, Phys. Rev. Lett. 75, 4130.
Loram, J.W., K.A. Mirza, J.R. Cooper, and W.Y. Jang, 1993, Phys. Rev. Lett. 71, 1740.
Lorenzo, J.E., J.M. Tranquada, D.J. Buttrey, and V. Sachan, 1995, Phys. Rev. B 51, 3176.
Louis, E., F. Guinea, M.P.L. Sancho, and J.A. Verges, 2001, Phys. Rev. B 64, 205108.
Löw, U., V.J. Emery, K. Fabricius, and S.A. Kivelson, 1994, Phys. Rev. Lett. 72, 1918.
Luther, A., and V.J. Emery, 1974, Phys. Rev. Lett. 33, 589.
MacLaughlin, D.E., 1976, Solid State Phys. 31, 2.
MacLaughlin, D.E., J.P. Vithayathil, H.B. Brom, J.C.J.M. de Rooy, P.C. Hammel, P.C.
Canfield, A.P. Reyes, Z. Fisk, J.D. Thomson, and S-W. Cheong, 1994, Phys. Rev. Lett. 72,
760.
202
Mahajan, A.V., H. Alloul, G. Collin, and J.F. Marucco, 1994, Phys. Rev. Lett. 72, 3100.
Mahajan, A.V., H. Alloul, G. Collin, and J.F. Marucco, 2000, Eur. Phys. J. B 13, 457.
Maki, K., and H. Won, 1994, Phys. Rev. Lett. 72, 1758.
Manske, D., I. Eremin, and K.H. Benneman, 2001, Phys. Rev. B 63, 054517.
Marshall, D.S., D.S. Dessau, A.G. Loeser, C-H. Park, A.Y. Matsuura, J.N. Eckstein, I.
Bozovic, P. Fournier, A. Kapitulnik, W.E. Spicer, and Z.-X. Shen, 1996, Phys. Rev. Lett.
76, 4841.
Marston, J.B., J.O. Fjaerestad, and A. Sudbö, 2002,Rev. Lett. 89, 056404.
Martin, I., G. Ortiz, A.V. Balatsky, and A.R. Bishop, 2001, Europhys. Lett. 56, 849.
Martin, I., A.V. Balatsky, and J. Zaanen, 2002, Phys. Rev. Lett. 88, 097003.
Martins, G.B., C. Gazza, J.C. Xavier, A. Feiguin, and E. Dagotto, 2000, Phys. Rev. Lett.
84, 5844.
Martins, G.B., J.C. Xavier, C. Gazza, M. Vojta, and E. Dagotto, 2001, Phys. Rev. B 63,
014414.
Mason, T.E., A. Schröder, G. Aeppli, H.A. Mook, and S.M. Hayden, 1996, Phys. Rev. Lett.
77, 1604.
Matsuda, M., Y. Endoh, K. Yamada, H. Kojima, I. Tanake, R.J. Birgeneau, M.A. Kastner,
and G. Shirane, 1992, Phys. Rev. B. 45, 12548.
Matsumura, M., M. Mali, J. Ross, and D. Brinkmann, 1997, Phys. Rev. B 56, 8938.
Matveenko, S.I., and S.I. Mukhin, 2000, Phys. Rev. Lett. 84, 6066.
Mazin, I.I., and V.M. Yakovenko, 1995, Phys. Rev. Lett. 75, 4134.
McQueeney, R.J., J.L. Sarrao, and R. Osborn, 1999a, Phys. Rev. B 60, 80.
203
McQueeney, R.J., Y. Petrov, T. Egami, M. Yethiraj, G. Shirane, and Y. Endoh, 1999b,
Phys. Rev. Lett. 82, 628.
McQueeney, R.J., A.R. Bishop, Y.S. Yi, and Z.G. Yu, 2000, J.Phys.: Condens. Matter 12,
L317.
McQueeney, R.J., J.L. Sarrao, P.G. Pagliuso, P.W. Stephens, and R. Osborn, 2001, Phys.
Rev. Lett. 87, 077001.
Mehring, M., 1992, Appl. Magn. Reson. 3, 383.
Mehta, A., and P.J. Heaney, 1994, Phys. Rev. B 49, 563.
Meixner, S., W. Hanke, E. Demler, and S.C. Zhang, 1997, Phys. Rev. Lett. 79, 4902.
Mendels, P., J. Bobroff, G. Collin, H. Alloul, M. Gabay, J.F. Marucco, N. Blanchard, and
B. Grenier, 1999, Europhys. Lett. 46, 678.
Metzner, W., and D. Vollhardt, 1989, Phys. Rev. Lett. 62, 324.
Mila, F., and T.M. Rice, 1989, Physica C 157, 561.
Millis, A.J., H. Monien, and D. Pines, 1990, Phys. Rev. B 42, 167.
Millis, A.J., and H. Monien, 1992, Phys. Rev. B 45, 3059.
Millis, A.J., and H. Monien, 1996, Phys. Rev. B 54, 16172.
Mitrović, V.F., E.E. Sigmund, M. Eschrig, H.N. Bachman, W.P. Halperin, A.P. Reyes, P.
Kuhns, and W.G. Moulton, 2001, Nature 413, 501.
Molegraaf, H.J.A., C. Presura, D. van der Marel, P.H. Kes, and M. Li, 2002, Science 295,
2239.
Monien, H., D. Pines, and M. Takigawa, 1991, Phys. Rev. B 43, 258.
Monthoux, P., A.V. Balatsky, and D. Pines, 1991, Phys. Rev. Lett. 67, 3448.
204
Monthoux, P., and D. Pines, 1992, Phys. Rev. Lett. 69, 961.
Mook, H.A., M. Yethiraj, G. Aeppli, T.E. Mason, and T. Armstrong, 1993, Phys. Rev. Lett.
70, 3490.
Mook, H.A., Pengcheng Dai, F. Doğan, and R.D Hunt, 2000, Nature 404, 729.
Mook, H.A., Pengcheng Dai, and F. Doğan, 2001, Phys. Rev. B 64, 012502.
Mook, H.A., Pengcheng Dai, and F. Doğan, 2002, Phys. Rev. Lett. 88, 097004.
Moonen, J.T., and H.B. Brom, 1995, Physica C 244, 1 and 10.
Moriya, T., 1963, J. Phys. Soc. Jap. 18, 516.
Morr, D.K., and D. Pines, 1998, Phys. Rev. Lett. 81, 1086.
Morr, D.K., and D. Pines, 2000a, Phys. Rev. B 62, 15177.
Morr, D.K., and D. Pines, 2000b, Phys. Rev. B 61, R6483.
Mukhin, S.I., 2000, Phys. Rev. B. 62, 4332.
Nachumi, B., Y. Fudamoto, A. Keren, K.M. Kojima, M. Larkin, G.M. Luke, J. Merrin, O.
Tchernyshyov, Y.J. Uemura, N. Ichikawa, M. Goto, H. Takagi, S. Uchida, M.K. Crawford,
E.M. McCarron, D.E. MacLaughlin, and R.H. Heffner, 1998, Phys. Rev. B 58, 8760.
Nakajima, K., K. Yamada, S. Hosoya, Y. Endoh, M. Greven, and R.J. Birgeneau, 1995, Z.
Phys. B 96, 479.
Narath, A., 1967, Phys. Rev. 162, 333.
Nayak, C., and F. Wilczek, 1996, Int. J. Mod. Phys. B 10, 2125.
Nayak, C., and F. Wilczek, 1997, Phys. Rev. Lett. 78, 2465.
Niedermayer, Ch., C. Bernhard, T. Blasius, A. Golnik, A. Moodenbaugh, and J.I. Budnick,
1998, Phys. Rev. Lett. 80, 3843.
205
Niemoller, T., N. Ichikawa, T. Frello, H. Hünnefeld, N.H Andersen, S. Uchida, J.R. Schneider,
and J.M. Tranquada, 1999, Eur. Phys. J. B 12, 509.
Noda, T., H. Eisaki, and S.-I. Uchida, 1999, Science 286, 265.
Norman, M.R., and H. Ding, 1998, Phys. Rev. B 57, R11089.
Norman, M.R., 2001, Phys. Rev. B 63, 092509.
Normand, B., H. Kohno, and H. Fukuyama, 1995, J. Phys. Soc. Jpn. 64, 3903.
Nussinov, Z., J. Rudnick, S.A. Kivelson, and L.N. Chayes, 1999, Phys. Rev. Lett. 83, 472.
Odier, P., N.J. Poirot, P. Simon, and D. Desousa Meneses, 1999, Eur. Phys. J. AP 5, 123.
Onose, Y., Y. Taguchi, T. Ishikawa, S. Shinomori, K. Ishizaka, and Y. Tokura, 1999, Phys.
Rev. Lett. 82, 5120.
Onose, Y., Y. Taguchi, K. Ishizaka, and Y. Tokura, 1999, Phys. Rev. Lett. 87, 217001.
Onufrieva, F., and J. Rossat-Mignod, 1995, Phys. Rev. B 52, 7572.
Pan, S.H., E.W. Hudson, A.K. Gupta, K.-W. Ng, H. Eisaki, S. Uchida, and J.C. Davis, 2000,
Phys. Rev. Let. 85, 1536.
Pao, C.-H., and Bickers, N.E., 1995, Phys. Rev. B 51, 16310.
Pashkevich, Yu.G., V.A. Blinkin, V.P. Gnezdilov, V.S. Kurnosov, V.V. Tsapenko, V.V.
Eremenko, P. Lemmens, M. Fischer, M. Grove, G. Güntherodt, L. Degiorgi, P. Wachter,
J.M. Tranquada, and D.J. Buttrey, 2000a, Physica B 284, 1473.
Pashkevich, Yu.G., V.A. Blinkin, V.P. Gnezdilov, V.V. Tsapenko, V.V. Eremenko, P. Lemmens, M. Fischer, M. Grove, G. Güntherodt, L. Degiorgi, P. Wachter, J.M. Tranquada, and
D.J. Buttrey, 2000b, Phys. Rev. Lett. 84, 3919.
Pellegrin, E., J. Zaanen, H.-J. Lin, G. Meigs, C.T. Chen, H. Eisaki, and S. Uchida, 1996,
Phys. Rev. B 53, 10667.
206
Pennington, C.H., S. Yu, K.R. Gorny, M.J. Buoni, W.L. Hults, and J.L. Smith, 2001, Phys.
Rev. B 63, 054513.
Pines, D., 1997, Z. Phys. B 103, 129.
Poilblanc, D., and T.M. Rice, 1989, Phys. Rev. B 39, 9749.
Poirot, N.J., Ch. Allancon, P. Odier, P. Simon, J.M. Bassat, and J.P. Loup, 1998, J. Solid
State Chem. 138, 260.
Polkovnikov, A., S. Sachdev, and M. Vojta, 2001, Phys. Rev. Lett. 86, 296.
Polkovnikov, A., M. Vojta, and S. Sachdev, 2002, Phys. Rev. B 65, 220509.
Prelovs̆ek, and X. Zotos, 1993, Phys. Rev. B 47, 5984.
Prelovs̆ek, T. Tohyama, and S. Maekawa, 2001, Phys. Rev. B 64, 052512.
Pryadko, L.P., S.A. Kivelson, and D.W. Hone, 1998, Phys. Rev. Lett. 80, 5651.
Pryadko, L.P., S.A. Kivelson, V.J. Emery, Y.B. Bazaliy, and E.A. Demler, 1999, Phys. Rev.
B 60, 7541.
Rabello, S., H. Kohno, E. Demler, and S.-C. Zhang, 1998, Phys. Rev. Lett. 80, 3586.
Ramirez, A.P., P.L. Gammel, S-W. Cheong, D.J. Bishop, and P. Chandra, 1996, Phys. Rev.
Lett. 76, 447.
Read, N., and S. Sachdev, 1989, Phys. Rev. Lett. 62, 1694.
Read, N., and S. Sachdev, 1990, Phys. Rev. B 42, 4568.
Rice, D.E., and D.J. Buttrey, 1993, J. Solid State Chem. 105, 197.
Riera, J.A., 2001, Phys. Rev. B 64, 104520.
Rigamonti, A., F. Borsa, and P. Carretta, 1998, Rep. Prog. Phys. 61,1367 and references
therein.
207
Rokhsar, D.S., and S.A. Kivelson, 1988, Phys. Rev. Lett. 61, 2376.
Rossat Mignod, J., L.P. Regnault, C. Vettier, P. Burlet, J.Y. Henry, and G. Lapertot, 1991,
Physica B 169, 58.
Rossat Mignod, J., L.P. Regnault, C. Vettier, P.Bourges, P. Burlet, J. Bossy, J.Y. Henry,
and G. Lapertot, 1991, Physica C 185-189, 86.
Rossat Mignod, J., L.P. Regnault, C. Vettier, P. Bourges, P. Burlet, J. Bossy, J.Y. Henry,
and G. Lapertot, 1992, Physica B 180-181, 383.
Sachan, V., D.J. Buttrey, J.M. Tranquada, J.E. Lorenzo, and G. Shirane, 1995, Phys. Rev.
B 51, 12742.
Sachdev, S., C. Buragohain, and M. Vojta, 1999, Science 286, 2479.
Sachdev, S., and S-C. Zhang, 2002, Science 295, 452.
Sachdev, S., and O. Park, 2002, Ann. Physics 298, 58.
Sadori, A., and M. Grilli, 2000, Phys. Rev. Lett. 84, 5375.
Saini, N.L., H. Oyanagi, A. Lanzara, D. Di Castro, S. Agrestini, A. Bianconi, F. Nakamura,
and T. Fujita, 2001, Phys. Rev. B. 64, 132510.
Salkola, M.I., and J.R. Schrieffer, 1998, Phys. Rev. B 58, R5944.
Satake, M., K. Kobayashi, T. Mizokawa, A. Fujimori, T. Tanabe, T. Katsufuji, and Y.
Tokura, 2000, Phys. Rev. B. 61, 15515.
Sato T., T. Kamiyama, T. Takahashi, K. Kurahashi, and K. Yamada, 2001, Science 291,
1517.
Savici, A.T., Y. Fudamoto, I.M. Gat, M.I. Larkin, Y.J. Uemura, G.M. Luke, K.M. Kojima,
Y.S. Lee, M.A. Kastner, and R.J. Birgenau, 2000, Physica B 289-290, 338.
Scalapino, D.J., and S.R. White, 1998, Phys. Rev. B 58, 8222.
208
Scalapino, D.J., S.-C. Zhang, and W. Hanke, Phys. Rev. B 58, 443.
Schrieffer, J.R., 1988, Theory of Superconductivity (Addison-Wesley, Redwood City) 4th
printing.
Schrieffer, J.R., X.G. Wen, and S.C. Zhang, 1988a, Phys. Rev. Lett. 60, 944.
Schrieffer, J.R., X.G. Wen, and S.C. Zhang, 1989, Phys. Rev. B 39, 11663.
Schulz, H.J., 1989, J. Phys. I 50, 2833.
Schulz, H.J., 1990, Phys. Rev. Lett 64, 1445.
Seibold, C. Castellani, C. Di Castro C, and M. Grilli, 1998, Phys. Rev. B 58, 13506.
Senthil, T., and M.P.A. Fisher, 2000, Phys. Rev. B 62, 7850.
Shastry, B.S., 1989, Phys. Rev. Lett. 63, 1289.
Shastry, B.S., and E. Abrahams, 1994, Phys. Rev. Lett. 72, 1933.
Shen, Z.-X., P.J. White, D.L. Feng, C. Kim, G.D. Gu, H. Ikeda, R. Yoshizaki, and N.
Koshizuka, 1998, Science 280, 259.
Shraiman, B.I., and E.D. Siggia, 1989, Phys. Rev. Lett. 62, 1564.
Si, Q.M., Y.Y. Zha, K. Levin, and J.P. Lu, 1993, Phys. Rev. B 47, 9055.
Sidis, Y., P. Bourges, H.F. Fong, B. Keimer, L.P. Regnault, J. Bossy, A. Ivanov, B. Hennion,
P. Gautier-Picard, G. Collin, D.L. Millius, and I.A. Aksay, 2000, Phys. Rev. Lett. 84, 5900.
Sidis, Y., C. Ulrich, P. Bourges, C. Bernhard, C. Niedermayer, L.P. Regnault, N.H. Andersen,
and B. Keimer, 2001, Phys. Rev. Lett. 86, 4100.
Singer, P.M., A.W. Hunt, A.F. Cederstrm, and T. Imai , 1999, Phys. Rev. B 60, 15345.
Singer, P.M., A.W. Hunt, and T. Imai , 2002a, Phys. Rev. Lett. 88, 047602.
Singer, P.M., and T. Imai, 2002b, Phys. Rev. Lett. 88, 187601.
209
Singley, E.J., D.N. Basov, K. Kurahashi, T. Uefuji, and K. Yamada, 2001, Phys. Rev. B 64,
224503.
Skinta, J.A., M.-S. Kim, T.R. Lemberger, T. Greibe, and M. Naito, 2002, Phys. Rev. Lett.
88, 207005.
Slichter, C.P., 1991, Principles of Magnetic Resonance (Springer-Verlag, Berlin).
Smith, C.M., A.H. Castro Neto, and A.V. Balatsky, 2001, Phys. Rev. Lett. 87, 177010.
Sonier, J.E., J.H. Brewer, R.F. Kiefl, R.I. Miller, G.D. Morris, C.E. Stronach, J.S. Gardner,
S.R. Dunsiger, D.A. Bonn, W.N. Hardy, R. Liang, and R.H. Heffner, 2001, Science 292,
1692.
Stemmann, G., C. Pépin, and M. Lavagna, 1994, Phys. Rev. B 50, 4075.
Stojkovic, B.P., Z.G. Yu, A.R. Bishop, A.H. Castro-Neto, and N. Gronbech-Jensen, 1999,
Phys. Rev. Lett. 82, 4679.
Suh, B.J., P.C. Hammel, Y. Yoshinari, J.D. Thompson, J.L. Sarrao, and Z. Fisk, 1998, Phys.
Rev. Lett. 81, 2791.
Suh, B.J., P.C. Hammel, M. Hücker, and B. Büchner, 1999, Phys. Rev. B 59, R3952.
Suh, B.J., P.C. Hammel, M. Hücker, B. Büchner, U. Ammerahl, and A. Revcolevschi, 2000,
Phys. Rev. B 61, R9265.
Sushkov, O.P., J. Oitmaa, and Z. Weihong, 2001, Phys. Rev. B 63, 104420.
Suzuki, T., T. Goto, K. Chiba, T. Shinoda, T. Fukase, H. Kimura, K. Yamada, M. Ohashi,
and Y. Yamaguchi, 1998, Phys. Rev. B. 57, R3229.
Takigawa, M., A.P. Reyes, P.C. Hammel, J.D. Thompson, R.H. Heffner, Z. Fisk, and K.C.
Ott, 1991, Phys. Rev. B 43, 247.
Takimoto, T., and T. Moriya, 1998, J. Phys. Soc. Jpn. 67, 3570.
210
Tallon, J.L., J.W. Loram, and G.V.M. Williams, 2002, Phys. Rev. Lett. 88, 059701.
Tamura, H., A. Hayashi, and Y. Ueda, 1996, Physica C 258, 61.
Tanamoto, T., K. Kuboki, and H. Fukuyama, 1991, J. Phys. Soc. Jpn. 60, 3072.
Tanamoto, T., H. Kohno, and H. Fukuyama, 1993, J. Phys. Soc. Jpn. 62, 717 and 1455.
Tanamoto, T., H. Kohno, and H. Fukuyama, 1994, J. Phys. Soc. Jpn. 63, 2739.
Tang, J.P., R.I. Dass, and A. Manthiram, 2000, Mater. Res. Bull. 35, 411.
Tchernyshyov, O., and L.P. Pryadko, 2000, Phys. Rev. B 61, 12503.
Tchernyshyov, O., M.R. Norman, and A.V. Chubukov, 2001, Phys. Rev. B 63, 144507.
Teitel’baum, G.B., I.M. Abu-Shiekah, O. Bakharev, H.B. Brom, and J. Zaanen, 2000, Phys.
Rev. B 63, 020507(R).
Teitel’baum, G.B., B. Büchner, and H. de Gronckel, 2000, Phys. Rev. Lett. 84, 2949.
Tewari, S., H-Y. Kee, C. Nayak, and S. Chakravarty, 2001, Phys. Rev. B 64, 224516.
Tinkham, M., 1975, Introduction to Superconductivity (Robert E. Krieger Publishing Company, Malabar).
Tranquada, J.M., K. Nakajima, M. Braden, L. Pintschovius, and R.J. McQueeney, 2002,
Phys. Rev. Lett. 88, 075505.
Tranquada, J.M., D.J. Buttrey, V. Sachan, and J.E. Lorenzo, 1994, Phys. Rev. Lett. 73,
1003.
Tranquada, J.M., B.J. Sternlieb, J.D. Axe, Y. Nakamura, and S. Uchida, 1995, Nature 375,
561.
Tranquada, J.M., J.D. Axe, N. Ichikawa, Y. Nakamura, S. Uchida, and B. Nachumi, 1996,
Phys. Rev. B 54, 7489.
211
Tranquada, J.M., P. Wochner, and D.J. Buttrey, 1997a, Phys. Rev. Lett. 79, 2133.
Tranquada, J.M., P. Wochner, A.R. Moodenbaugh, and D.J. Buttrey, 1997b, Phys. Rev. B
55, R6113.
Tranquada, J.M., N. Ichikawa, and S. Uchida, 1999, Phys. Rev. B 59, 14712.
Tsuei, C.C., and J.R. Kirtley, 2000, Phys. Rev. Lett. 85, 182.
Vajk, O.P., P.K. Mang. M. Greven, P.M. Gehring, and J.W. Lynn, 2002, Science 295, 1691.
van Bemmel, H.J.M., D.F.B. ten Haaf, W. van Saarloos, J.M.J. van Leeuwen, and G. An,
1994, Phys. Rev. Lett. 72, 2442.
van der Klink, J.J., and H.B. Brom, 2000, Progr. NMR Spectrosc. 36, 89.
van der Marel, D., A.J. Leggett, J.W. Loram, and J.R. Kirtley, 2002, Phys. Rev. B 66,
140501(R).
van Duin, C.N.A., and J. Zaanen, 2000, Phys. Rev. B 61, 3676.
Varma, C.M., P.B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, 1989,
Phys. Rev. Lett. 63, 1996.
Varma, C.M., 1999, Phys. Rev. Lett. 83, 3538.
Verges, J.A., E. Louis, P.S. Lomdahl, F. Guinea, and A.R. Bishop, 1991, Phys. Rev. B 43,
6099.
Viertio, H.E., and T. M. Rice, 1994, J. Phys.: Condens. Matter 6, 7091.
Vigliante, A., M. von Zimmermann, J.R. Schneider, T. Frello, N.H. Andersen, J. Madsen,
D.J. Buttrey, Doon Gibbs, and J.M. Tranquada, 1997, Phys. Rev. B 56, 8248.
Vojta, M., and S. Sachdev, 1999, Phys. Rev. Lett. 83, 3916.
Vojta, M., Y. Zhang, and S. Sachdev, 2000, Phys. Rev. B 62, 6721.
212
Wada, S., Y. Furukawa, M. Kaburagi, T. Kajitani, S. Hosoya, and Y. Yamada, 1993, J.
Phys.: Condens Matter 5,765.
Wakimoto, S., G. Shirane, Y. Endoh, K. Hirota, S. Ueki, K. Yamada, R.J. Birgeneau, M.A.
Kastner, P.M. Gehring, and S.H. Lee, 1999, Phys. Rev. B. 60, R769.
Wakimoto, S., R.J. Birgeneau, M.A. Kastner, Y.S. Lee, R. Erwin, P.M. Gehring, S.-H. Lee,
M. Fujita, K. Yamada, Y. Endoh, K. Hirota, and G. Shirane, 2000, Phys. Rev. B 61, 3699.
Wakimoto, S., R.J. Birgeneau, Y.S. Lee, and G. Shirane, 2001a, Phys. Rev. B. 63, 172501.
Wakimoto, S., J.M. Tranquada, T. Ono, K.M. Kojima, S. Uchida, S.-H. Lee, P.M. Geering,
and R.J. Birgeneau, 2001b, Phys. Rev. B. 64, 174505.
Walstedt, R.E., and S-W. Cheong, 1995, Phys. Rev. B 51, 3163.
Warren Jr., W.W., R.E. Walstedt, G.F. Brennert, R.J. Cava, R.F. Bell, and G. Dabbagh,
1989, Phys. Rev. Lett. 62, 1193.
Watanabe, I., M. Aoyama, M. Akoshima, T. Kawamata, T. Adachi, Y. Koike, S. Ohira, W.
Higemoto, and K. Nagamine, 2000a, Phys. Rev. B 62, R11985.
Watanabe, I., M. Aoyama, Y. Koike, S. Ohira, and K. Nagamine, 2000b, Phys. Rev. B 62,
14524.
Weger, M., 1972, Pure Appl. Chem. 32, 325.
Wen, X.G., and P. A. Lee, 1996, Phys. Rev. Lett. 76, 503.
Westfahl,H., J. Schmalian, and P.G. Wolynes, 2001, Phys Rev. B 6417, 4203.
White, S.R., 1992, Phys. Rev. Lett. 69, 2863.
White, S.R., and D.J Scalapino, 1998a, Phys. Rev. Lett. 80, 1272.
White, S.R., and D.J Scalapino, 1998b, Phys. Rev. Lett. 81, 3227.
213
White, S.R., and D.J Scalapino, 1999, Phys. Rev. B. 60, R753.
White, S.R., and D.J Scalapino, 2000a, Phys. Rev. B. 61, 6320.
White, S.R., and D.J Scalapino, 2000b, cond-mat/0006071.
White, S.R., I. Affleck, and D.J. Scalapino, 2002, Phys. Rev. B 65, 165122.
Williams, G.V.M., J.L. Tallon, E.M. Haines, R. Michalak, and R. Dupree, 1997, Phys. Rev.
Lett. 78, 721.
Winter, J., 1970, Magnetic Resonance in Metals (Oxford University Press, Oxford).
Wochner, P., J.M. Tranquada, D.J. Buttrey, and V. Sachan, 1998, Phys. Rev. B 57, 1066.
Yamada, K., C.H. Lee, K. Kurahashi, J. Wada, S. Wakimoto, S. Ueki, H. Kimura, Y. Endoh,
S. Hosoya, G. Shirane, R.J. Birgeneau, M. Greven, M.A. Kastner, and Y.J. Kim, 1998, Phys.
Rev. B 57, 6165.
Yamada, K., K. Kurahashi, Y. Endoh, R.J. Birgeneau, and G. Shirane, 1999, J. Phys. Chem.
Sol. 60, 1025.
Yamamoto, K., T. Katsufuji, T. Tanabe, and Y. Tokura, 1998, Phys. Rev. Lett. 80, 1493.
Yanagisawa, T., S. Koike, M. Miyazaki, and K. Yamaji, 2002, J. Phys. Chem. Sol. 63, 1379.
Yi, Y.S., Z.G. Yu, A.R. Bishop, and J.T. Gammel, 1998, Phys. Rev. B 58, 503.
Yin, L., S. Chakravarty, and P.W. Anderson, 1997, Phys. Rev. Lett. 78, 3559.
Yoshinari, Y., P.C. Hammel, and S.-W. Cheong, 1999, Phys. Rev. Lett. 82, 3536.
Yoshizawa, H., T. Kekeshita, R. Kajimoto, T. Tanabe, T. Katsufuji, and Y. Tokura, 2000,
Phys. Rev. B 61, R854.
Zaanen, J., G.A. Sawatzky, and J.W. Allen, 1985, Phys. Rev. Lett. 55, 418.
Zaanen, J., and O. Gunnarson, 1989, Phys. Rev. B 40, 7391.
214
Zaanen, J., and P.B. Littlewood, 1994, Phys. Rev. B 50, 7222.
Zaanen, J., and A.M. Oleś, 1996, Ann. Phys.-Leipzig 5, 224.
Zaanen, J., and W. van Saarloos, 1997, Physica C 282, 178.
Zaanen, J., 1998, J. Phys. Chem. Sol. 59, 1769.
Zaanen, J., O.Y. Osman, and W. van Saarloos, 1998, Phys. Rev. B 58, R11868.
Zaanen, J., 1999a, Science 286, 251.
Zaanen, J., 1999b, Physica C 318, 217.
Zaanen, J., 2000a, Phys. Rev. Lett. 84, 753.
Zaanen, J., 2000b, Nature 404, 714.
Zaanen, J., O.Y. Osman, H.V. Kruis, Z. Nussinov, and J. Tworzydlo, 2001, Phil. Mag. B
81, 1485.
Zaanen, J., 2002, Nature 415, 379.
Zachar, O., 2000, Phys. Rev. B. 65, 174411.
Zasadzinski, J.F., L. Ozyuzer, N. Miyakawa, K.E. Gray, D.G. Hinks, and C. Kendziora,
2001, Phys. Rev. Lett. 87, 067005.
Zha, Y., K. Levin, and Q. Si, 1993, Phys. Rev. B 47, 9124.
Zha, Y., V. Barzykin, and D. Pines, 1996, Phys. Rev. B 54, 7561.
Zhang, F.C., and T.M. Rice, 1988, Phys. Rev. B 37, 3759.
Zhang, S.-C., 1997, Science 275, 1089.
Zhang, S.-C., J.P. Hu, E. Arrigoni, W. Hanke, and A. Auerbach, 1997, Phys. Rev. B 60,
13070.
215
Zhang, Y., E. Demler, and S. Sachdev, 2002, Phys. Rev. B 66, 094501.
Zhou, X.J., P.V. Bogdanov, S.A. Keller, T. Noda, H. Eisaki, S. Uchida, Z. Hussain, and
Z.-X. Shen, 1999, Nature 286, 268.
Zhou, X.J., T. Yoshida, S.A. Kellar, P.V. Bogdanov, E.D. Lu, A. Lanzara, M. Nakamura, T.
Noda, T. Kakeshita, H. Eisaki, S. Uchida, A. Fujiimori, Z. Hussain, and Z.-X. Shen, 2001,
Phys. Rev. Lett 86, 5578.
Zhu, J.-X., I. Martin, and A.R. Bishop, 2002, Phys. Rev. Lett. 89, 067003.
Zimmermann, M.v., A. Vigliante, T. Niemöller, N. Ichikawa, T. Frello, J. Madsen, P.
Wochner, S. Uchida, N.H. Andersen, J.M. Tranquada, D. Gibbs, and J.R. Schneider, 1998,
Europhys. Lett. 41, 629.
216