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Magnetic neutron scattering in hole doped cuprate superconductors

A review is presented of the static and dynamic magnetic properties of hole-doped cuprate superconductors measured with neutron scattering. A wide variety of experiments are described with emphasis on the monolayer La_{2-x}(Sr,Ba)_{x}CuO_{4} and bilayer YBa_{2}Cu_{3}O_{6+x} cuprates. At zero hole doping, both classes of materials are antiferromagnetic insulators with large superexchange constants of J>100 meV. For increasing hole doping, the cuprates become superconducting at a critical hole concentration of x_{c}=0.055. The development of new instrumentation at neutron beam sources coupled with the improvement in materials has lead to a better understanding of these materials and the underlying spin dynamics over a broad range of hole dopings. We will describe how the spin dispersion changes across the insulating to superconducting boundary as well as the static magnetic properties which are directly coupled with the superconductivity. Experiments directly probing the competing magnetic and superconducting order parameters involving magnetic fields, impurity doping, and structural order will be examined. Correlations between superconductivity and magnetism will also be discussed.


Introduction
Magnetism plays an important role in the phase diagram of the high-temperature superconducting cuprates. A complete understanding of high-temperature superconductivity must involve a thorough characterization and knowledge of the spin response across the entire phase diagram. 1,2 The magnetic properties throughout the phase diagram have proven to be immensely complex and sensitive to effects of disorder. Currently, there is no complete theory of the spin spectrum able to describe the measured response throughout the entire phase diagram it is also not understood how the magnetism relates to the superconducting properties.
The cuprate compounds at zero charge doping are antiferromagnetic insulators which consist of twodimensional planes of magnetic CuO 2 plaquettes coupled through a strong superexchange interaction. 3 With one hole per Copper atom, one would expect that the undoped cuprate would be metallic and many calculations have indeed made this prediction. However, all undoped cuprates are antiferromagnetic insulators with large Néel temperatures, illustrating the importance that electronelectron interactions play in the phase diagram. Hightemperature superconductivity is introduced in these materials through the doping of holes into the CuO 2 planes which causes a dramatic suppression of the longranged antiferromagnetism and at a critical hole doping results in a superconducting state. For very large hole dopings, superconductivity is suppressed in favor of a metallic ground state well described by Fermi-liquid theory.
The simplest high-temperature superconductor is based on monolayered La 2 CuO 4 . The phase diagram is illustrated in Fig. 1. The phase diagram is generic amongst all hole doped cuprates and La 2 CuO 4 doped with Stron- tium is chosen as an illustrative example. The hole doped phase diagram is similar to that of electron doping, though the later class of materials will not be described in this review. La 2 CuO 4 can be driven into the supercon-ducting phase by replacing La with Sr (La 2−x Sr x CuO 4 ) or by doping with excess Oxygens to form La 2 CuO 4+y . La 2 CuO 4+y allows the opportunity to study superconductivity in a stoichiometric crystal, free of structural defects. Superconductivity appears at hole dopings above 5.5%, reaching a maximum T c of ∼ 40 K at approximately 15% hole doping. The monolayer cuprates have been investigated in great detail throughout the underdoped phase diagram with much attention focused on characterizing the transition from an antiferromagnetic insulator to a superconductor. The other commonly studied cuprate is the bilayer YBa 2 Cu 3 O 6 system which has a maximum T c of 93 K. This material has two weakly coupled CuO 2 planes in each unit cell and is doped into the superconducting phase by adding oxygen atoms into the CuO chains in the basal plane of the unit cell. The presence of chains means that for particular oxygen concentrations, the oxygen atoms form a stochiometric superlattice. 4,5 This again allows the spin fluctuations to be investigated in the absence of structural disorder for particular oxygen concentrations throughout the phase diagram. The ability to dope holes while maintaining a well ordered structure has proven important in separating the effects of charge doping from those caused by the introduction of structural disorder into the crystal. We present a review of the magnetic properties of the hole-doped cuprate superconductors using magnetic neutron scattering. Emphasis is placed on the underdoped side of the phase diagram for concentrations below where T c is maximum. The paper is divided into four sections dealing with various aspects related to the superconductivity with emphasis on how the superconducting and magnetic order parameters are related. The first section examines the common magnetic dispersion measured throughout the cuprate phase diagram and shows a remarkably common underlying spin response amongst the hole-doped cuprates. The second section deals with the spin structure and how it changes on continuously doping holes from a Néel-ordered state to a high-temperature superconductor and how the magnetism is related to the presence of impurities. The third section deals with competing magnetic and superconducting order parameters as investigated through the tuning of structural order and the application of large magnetic fields. The fourth and final section of this review examines various common trends in the high-temperature cuprate phase diagram.

Universal dispersion in insulating and underdoped phases
Neutron inelastic scattering has played an important role in characterizing the cuprates, and in particular, in understanding the insulating and underdoped regions of the phase diagram. The low-energy magnetic fluctuations have been well investigated using triple-axis spectrometers at thermal and cold reactor sources. However, the large super-exchange interaction of the Cu 2+ spins has meant that a full understanding of the high-energy spin fluctuations, and hence the spin interactions, can only be obtained using new instrumentation at spallation neutron sources and accordingly there have been many recent developments towards understanding the spin prop- erties of the cuprates. This section describes the magnetic dispersion in the monolayer La 2−x (Sr,Ba) x CuO 4 (LSCO or LBCO) and bilayer YBa 2 Cu 3 O 6+x (YBCO) superconductors. It is divided into two segments dealing with the low-energy fluctuations investigated using cold and thermal neutron sources, and high-energy fluctuations probed using epithermal neutrons from spallation neutron labs.

Low-energy fluctuations
Low-energy fluctuations in insulating LSCO and YBCO show very similar behavior, illustrating a common antiferromagnetic ground state in the insulating cuprates. 6,7 Both insulating YBCO and La 2 CuO 4 are dominated by large nearest neighbor superexchange J > 100 meV and small in-and out-of-plane anisotropies. The excitations are well understood in terms of linear spin-wave theory. Upon hole doping, the commensurate antiferromagnetic order is destroyed in favor of an incommensurate spin-density wave characterized by magnetic peaks symmetrically displaced away from the (π,π) position. This has been investigated in detail over a broad range in energy transfers for a variety hole concentrations in LSCO between p=0.07-0.2. The magnetic excitations for energy transfers below ∼ 20 meV consist of nearly vertical rods extending from the incommensurate positions. 8-10 Further work on excess oxygen doped La 2 CuO 4+y has shown that the low-energy magnetic incommensurate scattering is present in the case of excess oxygen doping where effects due to structural disorder may be removed. 11 The incommensurate scattering at low-energies is illustrated in Fig. 2, which shows data from excess oxygen La 2 CuO 4.055 and La 1.9 Sr 0.1 CuO 4 at several energy transfers. This magnetic incommensurate scattering is directly coupled with the superconductivity as illustrated by two key experimental observations. First, the value for the incommensurate wave vector defining the displacement of the magnetic scattering away from the (π,π) position scales with the superconducting transition temperature. This property will be discussed in detail later. Second, for a broad range of hole dopings, the low-energy magnetic response is suppressed for temperatures below the superconducting transition temperature, indicating the formation of a spin gap. These two points suggest that the incommensurate magnetic scattering is directly associated with hightemperature superconductivity.
The temperature dependence of the magnetic scattering at finite energy transfers in the normal state has been shown to follow a simple ω/T scaling law over a very broad range of hole-dopings extending from the spin-glass to the optimally doped regions. [12][13][14] Scaling has also been observed in La 2 Cu 1−x Li x O 4 implying that this property is common among the doped monolayer cuprates. 15 The only assumption behind the presence of scaling is that the dominant energy scale is defined by the temperature and naturally this assumption is expected to break down at energies comparable with the anisotropic exchange terms. At low-energies, a breakdown of scaling has been observed and thought to be the result of the out of plane anisotropy in the spin exchange. 16 The low-energy excitations in the bilayer YBCO 6+x system have been studied in most detail for strongly superconducting samples. [17][18][19] Below T c , the excitation spectrum is dominated by a strong commensurate peak in the neutron cross-section. At optimal doping the peak is centered at 41 meV and was originally thought to define a spin-gap and to be associated with the pairing boson for superconductivity or with the presence of a pseudogap in the spin dynamics. 20,21 Recent studies on underdoped YBCO have found that there is indeed significant magnetic scattering below the resonance energy with a similar incommensurate and anisotropic lineshape and spectral weight to that measured in the monolayer LSCO. 19,[22][23][24][25] It has also been demonstrated through a number of experiments by many groups that the incommensurability in YBCO 6+x scales with the superconducting transition temperature in a similar manner to that carefully measured in the monolayer cuprates, 26 confirming that indeed the magnetic fluctuations are directly coupled with the superconductivity. The anisotropy of the momentum lineshape at low energies where the incommensurate peaks are predominately displaced along the [100] directions implies that an explanation in terms of a simple conventional Fermi-surface nesting is unlikely and therefore models involving stripes or spiral spin phases may be more appropriate. 57 This point is further emphasized through the study of Lee et al. on excess oxygen stage-4 LaCuO 4+y (T c =42 K) who found the magnetic peaks incommensurate with the lat-La 2 CuO 4 , T=295 K tice and hence not precisely aligned with the Cu-O-Cu tetragonal directions. 27 However, we note that recent calculations have shown that models based on transitions from a truncated Fermi surface may indeed explain the anisotropy observed in the cuprates. 28 Further work and experiments are required to distinguish between a quasiparticle or spin based model. The combined results on excess oxygen La 2 CuO 4+y , YBCO, and LSCO confirm that the incommensurate scattering is a common feature to all cuprates and is not unique to LSCO nor the monolayer cuprates. The magnetic fluctuations in the very underdoped region of the YBCO 6+x diagram have recently been studied in detail due to recent advances in materials preparation. 30,31 The results showed a similar energy lineshape in the inelastic channel to that observed in very underdoped LSCO. 29 The temperature dependence of the low-energy fluctuations in doped YBCO 6+x is also well described by an ω/T law as is the case in LSCO. 32 It is interesting to note that scaling is followed in both the YBCO 6+x and LSCO cuprates over a very broad range of hole dopings implying that the origin of scaling is not simply the close proximity of a quantum critical point.

High-energy spin fluctuations
Measurements of the complete dispersion across the entire Brillouin zone provides a direct measure of the spin Hamiltonian. While measuring the low-energy spin response and the velocity of spin-waves gives the strength of nearest neighbor interactions, it is only a complete measure of the dispersion near the magnetic zone boundary that provides definitive information on the strength of higher order interactions. Measurements of the lowenergy spin response on single layer La 2 CuO 4 and bilayer YBa 2 Cu 3 O 6 were first conducted using thermal tripleaxis spectrometers but were limited in the energy range which could be covered. High energy measurements on LSCO and YBCO were first conducted on hot reactor neutron sources. 34 These measurements were able to observe successfully the spinwave velocity and to observe the presence of an optic mode in the bilayer YBCO system. Definitive studies of the spin-wave velocity were later made use spallation neutrons which allowed measurements in excess of 100 meV in energy transfer to be conducted. Early measurements by Hayden et al. at ISIS on La 2 CuO 4 and YBa 2 Cu 3 O 6 found nearest neighbor exchange constants of about 125 meV, consistent with triple-axis work done at low-energies. However, these early studies were not able to obtain data near the magnetic zone boundary and it was only more recently that Coldea et al. were able to measure the dispersion in La 2 CuO 4 across the entire Brillouin zone. 37 These experiments found the dispersion around the magnetic zone boundary to be anisotropic and characteristic of a ferromagnetic nextnearest exchange. An alternate explanation in terms of a ring exchange term was postulated. The spin-wave dispersion and intensity as a function of momentum transfer throughout the magnetic zone is shown in Fig. 3. The solid lines are calculations based on linear spin-wave theory with nearest neighbor exchange interactions of J=112 ± 4 and next-nearest J ′ =-11 ± 3 meV.
Measurements of the high-energy dispersion in the superconducting state were first conducted by Hayden et al. with later experiments studying the dispersion in YBCO by Arai et al. 35,36,38 Further work on LBCO, YBCO, and LSCO was conducted using improved instrumentation on the MAPS spectrometer at the ISIS spallation source. [39][40][41][42] The dispersion and integrated intensity as a function of energy transfer is plotted in Fig.  4 for superconducting Ortho-II YBCO 6.5 and in Fig. 5 for LBCO. The magnetic response in YBCO consists of both optic and acoustic modes resulting from the fact that it is a bilayer system with two Cu 2+ ions in each unit cell. In our discussion we will focus on the acoustic mode only. The data for the acoustic scattering show two energy scales, a low energy region where the dispersion is characterized by incommensurate scattering with an almost infinite slope and a high-energy scale where the excitations disperse with increasing energy. Measurements on optimally doped YBCO 6.85 have been conducted by Pailhes et al. 43 and show a very similar dispersion to measurements conducted in the underdoped region of the phase diagram. The experiments focused on the high-energy excitations have provided much new information on the spin spectrum and helped to clarify previous apparent differences between the bilayer YBCO and monolayer LSCO systems. The measurements conducted on LBCO and LSCO show that the low-energy incommensurate scattering do meet at a resonance energy which in LBCO is at about 50 meV. This is larger than the energy of the resonance peak in the YBCO cuprates and shows that the resonance peak is not directly related to the pairing boson in the cuprates as it does not scale with the critical temperature. The resonance also has too little spectral weight to be associated with the pairing boson as discussed by Kee et al., this is illustrated in Fig. 4 which shows that the resonance only makes up a small fraction of the total spectral weight. 44 These measurements conducted on several different cuprates over a broad range of superconducting hole dopings show a remarkably consistent dispersion with two distinct energy scales and the resonance defining a cross-over point (Figs. 4 and 5). The low-energies are characterized by incommensurate scattering with an anisotropic lineshape in momentum which meet at the resonance energy. At higher energies, the scattering disperses in a manner similar to that measured in the insulating cuprates with a much reduced spin-wave velocity indicative of a significant weakening of the nearest neighbor super-exchange. Whether or not the momentum lineshape at high-energies is closer to a circle or square is currently not resolved. However, Tranquada et al. 41 and Stock et al. 39 have shown that the spectral weight at high-energy transfers (Figs. 4 and 5) is similar to that measured in the insulator and therefore the high-energy spin excitations appears to be more consistent with spinwave type excitations. The recent measurements on the high-energy dispersion of the cuprates have revealed a general dispersion which is common among the monolayer and bilayers systems and possibly to all cuprates. It will be important for new theories to explain and model this and to explain its difference from the behavior in the insulator.

Discovery of diagonal spin density modulation
In the previous section the underlying spin dispersion was discussed throughout the underdoped region of the phase diagram. The magnetic spectrum in the superconducting state is clearly very different to that of the antiferromagnetic insulator. The incommensurability, which defines the low-energy spectrum, is directly related to the superconductivity and scales simply with the doping. In this section, the transition from an antiferromagnetic insulator to a superconductor with incommensurate magnetic fluctuations is discussed in detail and the origin of the incommensurate ordering is examined through impurity doping.
The linear relation δ = x in the underdoped superconducting (SC) phase 45 and the sharp incommensurate (IC) elastic peaks observed in the vicinity of x=0.12 46 triggered a systematic exploration of spin correlations in the spin-glass (SG) phase. How does the incommensurability disappear towards the undoped insulating phase? Does the sharp IC elastic peaks exist in the wider doping region down to the SG phase?
In order to answer these questions Wakimoto et al. revisited the spin correlations in the SG phase near the SC-SG boundary where they discovered a new type of spin density modulation by elastic scattering. 47 We call it a diagonal spin density modulation (D-SDM) because of the IC peak positions are rotated by 45 • in reciprocal space about (π,π) from those observed in the SC phase. Therefore, the modulation runs along the direction diagonal to the Cu-O bonds in CuO 2 planes, in contrast to the direction in the SC phase which is parallel/perpendicular to the Cu-O bonds. For the latter, we call the modulation in the SC phase a parallel spin density modulation (P-SDM) to distinguish the two types of spin modulation. One more important discovery by Wakimoto et al. 48 was the one-dimensionality of the D-SDM. In general, due to the orthorhombic symmetry, crystals contain multiple domains, however, they were fortunate to study a crystal with just two domains, A and B. A single pair of IC peaks was observed, and it was possible to uniquely associate them with domain A. As shown in Fig. 6, it turned out that the direction of the modulation is the b-axis, which is parallel to the buckling direction of the CuO 6 octahedra Following these experimental results, Matsuda et al. 49 performed a systematic neutron scattering study on the doping and temperature dependences of the D-SDM. They observed that the IC elastic peaks appear at low temperatures throughout the SG phase. From scans along the [001] direction which determines the L- dependence of the peak intensities, they further confirmed that the spin correlations are predominantly confined within the CuO 2 planes, with the Cu 2+ spins weakly coupled between nearest-neighbor planes.
Interestingly, the observed incommensurability in the SG phase approximately maps onto the linear relation between δ and x observed in the SC phase. Based on a simple charge stripe model, this doping dependence suggests that the charge density per unit length of a stripe is almost constant throughout the phase diagram, even when the modulation rotates away by 45 • near the SC boundary. However, detailed measurements show that there exists a deviation downwards from the linear relation towards the antiferromagnetic (AFM) phase. In fact as shown in Fig. 7, at the lowest values for x the density deviates towards 1 hole/Cu instead of 0.5 hole/Cu as in the parallel charge stripes.
How does the D-SDM disappear in the AF phase with x < 0.02? Matsuda et al. 50 examined the doped AFM phase by a high resolution neutron scattering measurement. Surprisingly, the D-SDM starts to develop below the Néel temperature and coexists with the 3D-AFM order. The incommensurability, however, does not depend on x. Instead, the IC magnetic peak intensity from the D-SDM phase decreases with decreasing x. Furthermore, upon cooling the samples, the IC elastic peaks appear simultaneously with the reduction in the commensurate magnetic Bragg peak intensities (Fig. 8). Therefore, the short ranged D-SDM phase precipitates from the AFM phase at low temperature, which suggests that electronic phase separation of the doped holes occurs so that some regions with hole concentration p ∼ 0.02 and the rest with p∼ 0 are formed. It is noted that the reduction of the magnetic Bragg peak intensity upon cooling as shown in Fig. 8 was observed by Endoh et al. in LCO nearly 20 years ago. 51 We now finally understand the origin of such What is the direct relevance of the diagonal-parallel SDM transition to the superconductivity? In order to answer this question Fujita et al. 52 prepared single crystals of LSCO with finely tuned hole concentrations near the boundary. Uniform magnetic susceptibility measurements for these single crystals revealed that the SC transition sharply disappears below x c ∼ 0.055. Then a detailed neutron scattering study using crystals with x above and below x c finally confirmed the coincidence between the appearance of the P-SDM and the superconductivity. However, in the SC phase near the boundary the D-SDM and P-SDM with similar incommensurabilities coexist as demonstrated by a circular-scan around (π,π). This coexistence suggests a doping-induced first order transition between the diagonal and parallel SDM phases and between the SG and SC phases. It is noted that a µSR experiment observed the penetration of the SG phase into a wide region of the underdoped SC phase. 53 We speculate that the magnetic signal observed by µSR corresponds to the elastic magnetic signal from the P-SDM in the underdoped SC phase.
What is the microscopic origin of the D-SDM? As in the case of the P-SDM, one possibility to explain the D-SDM is the charge stripe model. The parallel stripe phase in La 2−x−y Nd y Sr x CuO 4 and La 2−x Ba x CuO 4 is stabilized by the structural distortion, namely, the low temperature tetragonal (LTT) structure, which is strongly coupled to the charge ordering. 54,55 However, in lightly hole-doped LSCO, which has no LTT phase, no evidence has been found that the charge stripe phase is realized. Most importantly, the structural distortion, originating from the charge ordering, has not been observed, al- though this non-observation can be due to disorder in the periodicity and direction of the stripe. On the other hand, Hasselman et al. 56,57 reported that the D-SDM may be explained using the spiral spin model, originating from the magnetic frustration around the localized hole spins. In this model, the IC magnetic peaks are purely magnetic in origin and have nothing to do with charge ordering. Matsuda et al. 58 studied the effects of impurities to differentiate between different models for the D-SDM. The results show that Zn doping reduces the incommensurability just slightly. On the other hand, Ni doping quickly destroys the incommensurability and restores the Néel ordering, indicating a strong effect on hole localization. This suggests that Ni is doped as Ni 3+ or as Ni 2+ with a hole forming a strongly bound state. Therefore, Ni doping reduces the number of mobile or hopping Zhang-Rice (ZR) singlet states around the Cu spins by creating localized hole sites near the doped Ni. Then the concentration of the mobile ZR singlet x ef f can be described by the difference between the number of holes and doped Ni ions. In fact, the incommensurability of the D-SDM for the Ni doped samples can be summarized by x ef f as shown in Fig. 10. Furthermore, the onset temperatures of the D-SDM in Ni doped samples correspond to those of Ni free samples with hole concentration x ef f . This means that the incommensurability in this system is controlled by the number of mobile ZR singlets or mobile holes rather than the number of localized holes around Ni. Furthermore, it should be noted that due to the strong localization the effective spin value near the doped Ni is expected to be 1/2, the same as the surrounding Cu spins.
Hence the magnetic effect of Ni doping should be smaller than that of Zn in contradiction to the experimental result. Therefore, the observed impurity effect on the spin density modulation, particularly the Ni effect favors the charge stripe model based on charge segregation by mobile holes rather than the spiral spin model based on localized hole spins. It is noted that the strong localization effect around Ni impurities is also observed in the lightly doped AFM phase, where the Néel temperature quickly recovers and the precipitation of the D-SDM is suppressed by Ni doping. 59 The comparison between doping holes and impurities directly on the Cu 2+ sites reveals a strong connection between antiferromagnetism and superconductivity. These competing order parameters are further discussed in the next section by tuning the structural order through annealing and the superconducting order parameter directly by applying a magnetic field.

Competing Superconducting and Magnetic Order
As discussed in the previous three sections, the Copper oxide high temperature superconductors exhibit remarkably rich and complicated static and dynamic spin fluctuation phenomena across the entire phase diagram. 1 It is particularly striking that in underdoped La 2−x Sr x CuO 4 and related compounds static incommensurate magnetic order and superconductivity coexist. It is found further that for optimally doped La 2 CuO 4.11 , which is a stoichiometric ordered, single crystal phase without appreciable structural disorder, the superconducting and magnetic phase transitions occur at the same temperature. 8 It is important to know whether or not these two kinds of order compete or cooperate with one another and whether they coexist microscopically or form spatially separate phases.
As is well known, the superconducting order may be diminished by either the introduction of structural disorder or by the application of a magnetic field which in type II superconductors in turn introduces vortices into the superconductor. Recently, a number of neutron scattering experiments have been carried out which probe the superconductivity and magnetic correlations in the presence of structural disorder and/or magnetic fields applied perpendicular to the CuO 2 planes. 60-68 These experiments have in turn stimulated important theoretical developments, most notably by Demler et al., [69][70][71][72] which have elucidated the nature of the competition between the incommensurate magnetic order and the superconductivity. We will discuss the predictions of these theories in the context of the experiments. We should emphasize that in this section we will not attempt a comprehensive review of this growing subfield of high temperature superconductivity but rather we shall give a few illustrative examples to evince the basic phenomenology and the essential theoretical ideas.
The first experiments to probe these effects were the studies by Yamada and coworkers of La 1.88 Sr 0.12 CuO 4 (T c =12 K) in a magnetic field. 61 However, since microscopic structural disorder is a feature of nearly all high temperature superconductors, it is convenient to review Stage-4 La 2 CuO 4.11 , E=0 meV first the work of Y.S. Lee et al. 60 probing the effects of disorder. Generally, it is difficult to isolate the effects of disorder on any phase transition, especially one with competing order parameters. However, as discovered by Lee et al., 60 stage-4 La 2 CuO 4.11 is a system in which the intercalated oxygen atoms can be progressively disordered by varying the annealing conditions. Specifically, for samples annealed at 320 K or lower the intercalated oxygen atoms order three dimensionally so that the material is a true stoichiometric single crystal. In such samples the superconducting transition temperature is ∼ 42 K (onset). Incommensurate magnetic order with incommensurability of ∼ 1/8 occurs at precisely the same temperature. 27 By annealing the sample at temperatures of 330 K or higher and then quenching to low temperatures the oxygen lattice can be completely disordered. Measurements of the diamagnetic susceptibility show that for an anneal at 360 K and subsequent quench to 9 K T c is reduced by 5 K to 37 K. We show in Fig. 11 A the intensity of the spin density wave (SDW) peak as a function of temperature in the oxygen ordered state versus the oxygen disordered state . Fig 11 B shows data at H = 0 T and H =7.5 T. We will discuss these latter data later in this section. As is evident from Fig. 11 A the effect of disordering the intercalant oxygen atoms is to enhance the magnetic intensity by nearly a factor of 2 while reducing T c by about 5 K. None of the magnetic phase transition temperature, the magnetic wave vector or the magnetic correlation length is measurably affected. These results indicate that the superconductivity and the SDW order compete and that microscopic structural disorder favors the magnetic order over the superconductivity. The sensitivity of the SDW amplitude to structural disorder also explains the variability of the magnetic intensity in different samples of nominally the same composition. 27, 60-64 As noted above, the first experiments to probe the effects of a magnetic field on the magnetic and superconducting order were those of Katano et al. 61 on La 1.88 Sr 0.12 CuO 4 (T c =12 K). They found that a field of 10 T applied perpendicular to the CuO 2 planes severely depressed Tc while the low temperature magnetic intensity increased by as much as 50% suggesting that the superconductivity and SDW order compete. These experiments were followed by a comprehensive study by Lake et al. 62 of the low energy spin dynamics in the optimally doped material La 1.84 Sr 0.16 CuO 4 (T c =38.5 K) at H =0 T and H =7.5 T. This study showed, in addition to the expected suppression of T c , that a field of 7.5 T created new spin excitations in the superconducting spin gap. They related this explicitly to the antiferromagnetism in the cores of the vortices and they then made the prescient deduction that the spins in the vortices are correlated over a variety of length scales from the atomic to the mesoscopic.
Stimulated by these experiments, Demler et al. 69,70 proposed a Ginzburg-Landau model for coupled SC and SDW order. Their model built on earlier work by a number of authors. 2,71,72 They hypothesized that these data could be understood by assuming that the superconductor (SC) was in the vicinity of a bulk quantum phase transition to a state with microscopic coexistence of SC and SDW order. This led them to predict the phase diagram shown in Fig. 12. They further predicted that in the state with both SC and SW order, application of La 2 CuO 4+y , E=0 meV Essentially simultaneous with this theoretical development, Lake et al. 63 and Khaykovich et al. 64 measured the enhancement of the SDW intensity as a function of field in La 1.9 Sr 0.1 CuO 4 and stage-4 La 2 CuO 4.11 respectively. In both cases a significant enhancement of the SDW intensity is found. Importantly, the systems exhibit resolution limited magnetic diffraction peaks whose wave vectors are identical at high fields (H up to 14 T) to those of the zero field state. This necessitates that the induced magnetism has the same microscopic origin as that at zero field. Data in La 2 CuO 4.11 at H =0 T and 7.5 T are shown in Fig. 11 B. In both cases the intensity increases linearly with field as opposed to the H 2 dependence expected from a pure magnetic effect. Specifically, this singular dependence of the SDW amplitude on the applied field excludes purely magnetic mechanisms for the phenomenon such as a suppression of the fluctuations of the ordered moment by the applied field or an increase of the correlations along the c-axis each of which by symmetry must scale like H 2 . This result alone dictates that the SDW enhancement must be driven by the vortices whose density scales linearly with H. In fact, in both systems it is found that over the entire range of fields the intensity follows the Demler et al. prediction, ∆I ∼ H ln(aH/H c2 ) to within the experimental errors. Data for stage-6 and stage-4 La 2 CuO 4+y for fields from 0 to 14 T are shown in Fig. 13 together with the H ln(1/H) theoretical prediction.
Indirect confirmation of the model is given by the study by Matsuda et al. 65 of the effects of the appli-   Fig. 12, it should be possible to drive the system from the SC to the SC+SDW state with the application of a magnetic field and the transition should be second order. Consideration of all of the available neutron scattering data in the La 2−x Sr x CuO 4 system led Khaykovich et al. 67 to conclude that if Fig.  12 is correct then the QCP should lie near x ∼1/8. Accordingly, Khaykovich et al. studied the magnetic response as a function of magnetic field in a crystal with x=0.144, intermediate between the putative QCP and optimal doping. They confirmed earlier results that at H =0 T no static magnetic order is observable. However at fields above about 3 T a clear SDW signal corresponding to long range incommensurate order is observed. The detailed temperature dependence of the SDW diffraction signal is shown in Fig.14. The intensity is fitted according to a power law I(H) − I(BG) = A(H − H c ) β , with H c =2.7±0.8 T and β=0.36 ± 0.10. This experiment would seem to provide convincing evidence for the basic physics of the model of competing SC and SDW order proposed by Demler et al. 69,70 Further studies of both the static and dynamic magnetic behavior as a function of magnetic field around x=1/8 are required to complete the empirical picture and to determine definitively whether or not x=1/8 is indeed a QCP in the La2 − xSr x CuO 4 system.

Putting the pieces together
We have noted that the magnetic correlations in underdoped La 2−x Sr x CuO 4 are compatible with a model in which the spin incommensurability is a result of the segregation of the doped holes into stripes, forming antiphase domain walls. 2,[73][74][75][76] There is direct evidence for order of this kind in La 1.6−x Nd 0.4 Sr x CuO 4 and La 2−x Ba x CuO 4 (especially at x = 1 8 ) from diffraction measurements; 54,55,77,78 the charge order has recently been confirmed by soft-x-ray resonant diffraction. 79 By segregating the charge carriers, it is possible for magnetic domains to survive that are similar to those of the parent insulator. The superexchange interaction should still be operable in these domains, and the large energy scale for magnetic excitations found in La 2−x Sr x CuO 4 and La 2−x Ba x CuO 4 , as discussed in §2, is consistent with this picture. The similar dispersion of spin excitations in underdoped YBa 2 Cu 3 O 6+x suggests that the same sort of stripe-like correlations are present there as well.
The diagonal spin modulation found at light doping in La 2−x Sr x CuO 4 between the antiferromagnetic and superconducting phases might not be a universal feature among the cuprates. Studies of the bilayer system La 2−x (Sr,Ca) x CaCu 2 O 6+δ with x = 0.1-0.2 (corresponding to 5-10% hole doping) reveal patches of commensurate short-range antiferromagnetic order that survive to temperatures > 100 K, 80, 81 even though optical conductivity measurements demonstrate the presence of mobile holes in the CuO 2 planes. 82 The common feature seems to be that doped holes do not mix well with commensurate antiferromagnetism. The coexistence of distinct regions of antiferromagnetic order and of mo- bile holes in La 2−x (Sr,Ca) x CaCu 2 O 6+δ might reflect substantial local variations in chemical potential balanced by Coulomb interactions; in any case, the large electronic disorder in this system is correlated with a small superconducting volume fraction, as determined by magnetic susceptibility measurements. 81 Returning to the superconducting regime, the stripe picture is plausible in the under-doped region, where stripes can maintain a reasonable separation. If the stripes all have a similar hole concentration, then increasing the hole density leads to a rise in the stripe density. Experimentally, the magnetic incommensurability, which should be proportional to the stripe density, increases with doping, as shown in Fig. 15, until it saturates above x ∼ 1 8 . At x = 1 8 , the separation between charge stripes is approximately four lattice spacings; it may be difficult to increase the stripe density beyond this point, as the stripes would lose their definition at smaller separations.
It is informative to consider the behavior at large doping. Wakimoto and coworkers 83 have studied the magnetic scattering from La 2−x Sr x CuO 4 with 0.25 ≤ x ≤ 0.30. At low temperatures, a peak in the inelastic scattering is found at ∼ 6 meV; however, the magnitude of the imaginary part of the dynamic susceptibility, χ ′′ , decreases steadily to zero. As shown in Fig. 16, they have found that the decrease in the superconducting transition temperature, T c , is correlated with χ ′′ .
What happens to the signal? One possibility would be for it to move to higher energy. The big question is whether the weight remains at an energy scale of some tens of meV, consistent with fluctuating local moments, or whether it moves to electronic energy scales, on the  order of eV. To test this, new measurements at ISIS have looked for magnetic excitations up to ∼ 100 meV in La 2−x Sr x CuO 4 with x = 0.25 and 0.30. 84 They demonstrate a drastic reduction in magnetic signal for ω > 10 meV, with the signal at lower energies consistent with the triple-axis work. 83 Thus, the correlation shown in Fig. 16 truly shows that superconductivity and local antiferromagnetism disappear together. (Note that it is not practical to perform similar measurements on YBa 2 Cu 3 O 6+x because so far it has not been possible to achieve such high doping levels.) Muon spin-relaxation studies have shown, in several cuprate systems, that there is a rapid decrease in the fraction of normal-state charge carriers that participate in the superfluid density as the hole concentration is raised above 20%. 85,86 There have also been reports of percolative behavior in over-doped samples based on magnetization studies of the superconducting state. 87 Such observations have led to suggestions that over-doped cuprates are characterized by electronic phase separation. 86,87 If we take the phase-separation scenario seriously, then the fact that the residual magnetic scattering at high doping is incommensurate suggests that one of the phases has stripe correlations similar to the x = 1 8 phase, with the volume fraction of this phase decreasing with x, as indicated in the middle panel of Fig. 17. The other phase is presumably uniformly doped. The picture, then, is that as one increases the doping beyond x ∼ 1 8 , it becomes unfavorable to accommodate the additional holes in stripes; instead, patches of the uniformly-doped phase grow at the expense of the stripe phase. The maximum T c seems to occur in a mixed phase region dominated by the stripe phase.
New evidence for stripe correlations in cuprate superconductors comes from a neutron scattering study of the bond-stretching phonon mode. 88 In stripe-ordered samples, a sharp drop in the dispersion, together with a large increase in energy width, is observed at q = (0.25, 0, 0), in reciprocal lattice units. A very similar anomaly is observed in superconducting La 2−x Sr x CuO 4 and YBa 2 Cu 3 O 6+x ; however, the anomaly is absent in nonsuperconducting La 2 CuO 4 and La 1.7 Sr 0.3 CuO 4 . There is a strong circumstantial case that the anomaly is associated with stripes. Further support comes from the absence of such an anomaly in conventional calculations. 89 Figure 18 shows the experimental correlation between the superconducting transition temperature and the magnetic incommensurability. 45 The observed trend is consistent with the theoretical arguments that electronic inhomogeneity (as in the form of charge stripes) is important for achieving high-temperature superconductivity. 90 The field-induced magnetic order discussed in §4 indicates that even static stripes are close in energy to the superconducting state.
The perspective throughout this article has been one in which the cuprates are viewed as doped antiferromagnets, with the magnetic correlations observed by neutron scattering arising from local Cu moments coupled by superexchange. In this picture, the carriers tend to segregate to define the magnetic regions. There is, of course, an alternative view in which the magnetic scattering is attributed entirely to the charge carriers. It is possible to calculate the dynamic susceptilibility from conventional formulas for quasiparticles with residual magnetic or Coulomb interactions. [91][92][93][94][95] Such an approach is, perhaps, more natural when one comes at the problem from the over-doped side. In principle, there should be excitations coming both from Cu moments and from the carriers. The challenge for the future is to learn how these different contributions combine and interact.

Acknowledgment
Three of us had the chance to enjoy the excitement of working with Gen Shirane on magnetic neutron scattering studies of the cuprates at the High Flux Beam Reactor (HFBR). Even after the HFBR's last cycle, Gen continued to direct collaborative experiments at international facilities for another decade. We have all benefited from Gen's knowledge, experience, and leadership, and we are deeply grateful for the opportunities that we had to work with, and learn from, him. JMT