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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 à 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. 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