Orbital symmetry of charge-density-wave order in La1.875Ba0.125CuO4 and YBa2Cu3O6.67

Discovery of orbital ordering in Bi2Sr2CaCu2O8+x

The primordial ingredient of cuprate superconductivity is the CuO2 unit cell. Theories usually concentrate on the intra-atom Coulombic interactions dominating the 3d9 and 3d10 configurations of each copper ion. However, if Coulombic interactions also occur between electrons of the 2p6 orbitals of each planar oxygen atom, spontaneous orbital ordering may split their energy levels. This long-predicted intra-unit-cell symmetry breaking should generate an orbitally ordered phase, for which the charge transfer energy ε separating the 2p6 and 3d10 orbitals is distinct for the two oxygen atoms. Here we introduce sublattice-resolved ε(r) imaging to CuO2 studies and discover intra-unit-cell rotational symmetry breaking of ε(r). Spatially, this state is arranged in disordered Ising domains of orthogonally oriented orbital order bounded by dopant ions, and within whose domain walls low-energy electronic quadrupolar two-level systems occur. Overall, these data reveal a Q = 0 orbitally ordered state that splits the oxygen energy levels by ~50 meV, in underdoped CuO2.

Prevailing Charge Order in Overdoped La_{2-x}Sr_{x}CuO_{4} beyond the Superconducting Dome

The extremely overdoped cuprates are generally considered to be Fermi liquid metals without exotic orders, whereas the underdoped cuprates harbor intertwined states. Contrary to this conventional wisdom, using Cu L_{3}-edge and O K-edge resonant x-ray scattering, we reveal a charge order (CO) correlation in overdoped La_{2-x}Sr_{x}CuO_{4} (0.35≤x≤0.6) beyond the superconducting dome. This CO has a periodicity of ∼6 lattice units with correlation lengths of ∼20 lattice units. It shows similar in-plane momentum and polarization dependence and dispersive excitations as the CO of underdoped cuprates, but its maximum intensity differs along the c direction and persists up to 300 K. This CO correlation cannot be explained by the Fermi surface instability and its origin remains to be understood. Our results suggest that CO is prevailing in the overdoped metallic regime and requires a reassessment of the picture of overdoped cuprates as weakly correlated Fermi liquids.

Atomic-scale interpretation of the quantum oscillations in cuprate superconductors

Cuprate superconductors display unusual features in bothkspace and real space as the superconductivity is suppressed-a broken Fermi surface, charge density wave, and pseudogap. Contrarily, recent transport measurements on cuprates under high magnetic fields report quantum oscillations (QOs), which imply rather a usual Fermi liquid behavior. To settle the disagreement, we investigated Bi₂Sr₂CaCu₂O_8+δunder a magnetic field in an atomic scale. A particle-hole (p-h) asymmetrically dispersing density of states (DOSs) modulation was found at the vortices on a slightly underdoped sample, while on a highly underdoped sample, no trace of the vortex was found even at 13 T. However, a similar p-h asymmetric DOS modulation persisted in almost an entire field of view. From this observation, we infer an alternative explanation of the QO results by providing a unifying picture where the aforementioned seemingly conflicting evidence from angle-resolved photoemission spectroscopy, spectroscopic imaging scanning tunneling microscopy, and magneto-transport measurements can be understood solely in terms of the DOS modulations.

Monte Carlo study of cuprate superconductors in a four-bandd-pmodel: role of orbital degrees of freedom

Understanding the various competing phases in cuprate superconductors is a long-standing challenging problem. Recent studies have shown that orbital degrees of freedom, both Cue_gorbitals and Oporbitals, are a key ingredient for a unified understanding of cuprate superconductors, including the material dependence. Here we investigate a four-bandd-pmodel derived from the first-principles calculations with the variational Monte Carlo method, which allows us to elucidate competing phases on an equal footing. The obtained results can consistently explain the doping dependence of superconductivity, antiferromagnetic and stripe phases, phase separation in the underdoped region, and also novel magnetism in the heavily-overdoped region. The presence ofporbitals is critical to the charge-stripe features, which induce two types of stripe phases withs)-wave andd-wave bond stripe. On the other hand, the presence ofdz2orbital is indispensable to material dependence of the superconducting transition temperature (Tc), and enhances local magnetic moment as a source of novel magnetism in the heavily-overdoped region as well. These findings beyond one-band description could provide a major step toward a full explanation of unconventional normal state and highTcin cuprate supercondutors.

Charge order textures induced by non-linear couplings in a half-doped manganite

The self-organization of strongly interacting electrons into superlattice structures underlies the properties of many quantum materials. How these electrons arrange within the superlattice dictates what symmetries are broken and what ground states are stabilized. Here we show that cryogenic scanning transmission electron microscopy (cryo-STEM) enables direct mapping of local symmetries and order at the intra-unit-cell level in the model charge-ordered system Nd_1/2Sr_1/2MnO₃. In addition to imaging the prototypical site-centered charge order, we discover the nanoscale coexistence of an exotic intermediate state which mixes site and bond order and breaks inversion symmetry. We further show that nonlinear coupling of distinct lattice modes controls the selection between competing ground states. The results demonstrate the importance of lattice coupling for understanding and manipulating the character of electronic self-organization and that cryo-STEM can reveal local order in strongly correlated systems at the atomic scale.

In this paper, the authors demonstrate that cryogenic scanning transmission electron microscopy allows for the direct mapping of the local arrangements and symmetries of electronic order, providing a useful method for studying strongly correlated systems. They show this using the example of Nd1/2Sr1/2MnO3, a model charge ordered material.

Dual Orbital Degeneracy Lifting in a Strongly Correlated Electron System

The local structure of NaTiSi_{2}O_{6} is examined across its Ti-dimerization orbital-assisted Peierls transition at 210 K. An atomic pair distribution function approach evidences local symmetry breaking preexisting far above the transition. The analysis unravels that, on warming, the dimers evolve into a short range orbital degeneracy lifted (ODL) state of dual orbital character, persisting up to at least 490 K. The ODL state is correlated over the length scale spanning ∼6 sites of the Ti zigzag chains. Results imply that the ODL phenomenology extends to strongly correlated electron systems.

Orbital symmetries of charge density wave order in YBa2Cu3O6+x

Charge density wave (CDW) order has been shown to compete and coexist with superconductivity in underdoped cuprates. Theoretical proposals for the CDW order include an unconventional d-symmetry form factor CDW, evidence for which has emerged from measurements, including resonant soft x-ray scattering (RSXS) in YBa₂Cu₃O_6+x (YBCO). Here, we revisit RSXS measurements of the CDW symmetry in YBCO, using a variation in the measurement geometry to provide enhanced sensitivity to orbital symmetry. We show that the (0 0.31 L) CDW peak measured at the Cu L edge is dominated by an s form factor rather than a d form factor as was reported previously. In addition, by measuring both (0.31 0 L) and (0 0.31 L) peaks, we identify a pronounced difference in the orbital symmetry of the CDW order along the a and b axes, with the CDW along the a axis exhibiting orbital order in addition to charge order.

Multiorbital charge-density wave excitations and concomitant phonon anomalies in Bi2Sr2LaCuO6+δ

Charge-density waves (CDWs) are a ubiquitous form of electron density modulation in cuprate superconductors. Unveiling the nature of quasistatic CDWs and their dynamical excitations is crucial for understanding their origin––similar to the study of antiferromagnetism in cuprates. However, dynamical CDW excitations remain largely unexplored due to the limited availability of suitable experimental probes. Here, using resonant inelastic X-ray scattering, we observe dynamical CDW excitations in Bi₂Sr₂LaCuO_6+δ (Bi2201) superconductors through its interference with the lattice. The distinct anomalies of the bond-buckling and the bond-stretching phonons allow us to draw a clear picture of funnel-shaped dynamical CDW excitations in Bi2201. Our results of the interplay between CDWs and the phonon anomalies shed light on the nature of CDWs in cuprates.

Charge-density waves (CDWs) are ubiquitous in underdoped cuprate superconductors. As a modulation of the valence electron density, CDWs in hole-doped cuprates possess both Cu-3d and O-2p orbital character owing to the strong hybridization of these orbitals near the Fermi level. Here, we investigate underdoped Bi₂Sr_1.4La_0.6CuO_6+δ using resonant inelastic X-ray scattering (RIXS) and find that a short-range CDW exists at both Cu and O sublattices in the copper-oxide (CuO₂) planes with a comparable periodicity and correlation length. Furthermore, we uncover bond-stretching and bond-buckling phonon anomalies concomitant to the CDWs. Comparing to slightly overdoped Bi₂Sr_1.8La_0.2CuO_6+δ, where neither CDWs nor phonon anomalies appear, we highlight that a sharp intensity anomaly is induced in the proximity of the CDW wavevector (Q_CDW) for the bond-buckling phonon, in concert with the diffused intensity enhancement of the bond-stretching phonon at wavevectors much greater than Q_CDW. Our results provide a comprehensive picture of the quasistatic CDWs, their dispersive excitations, and associated electron-phonon anomalies, which are key for understanding the competing electronic instabilities in cuprates.

Charge ordering in superconducting copper oxides

Charge order has recently been identified as a leading competitor of high-temperature superconductivity in moderately doped cuprates. We provide a survey of universal and materials-specific aspects of this phenomenon, with emphasis on results obtained by scattering methods. In particular, we discuss the structure, periodicity, and stability range of the charge-ordered state, its response to various external perturbations, the influence of disorder, the coexistence and competition with superconductivity, as well as collective charge dynamics. In the context of this journal issue which honors Roger Cowley's legacy, we also discuss the connection of charge ordering with lattice vibrations and the central-peak phenomenon. We end the review with an outlook on research opportunities offered by new synthesis methods and experimental platforms, including cuprate thin films and superlattices.

Local orbital degeneracy lifting as a precursor to an orbital-selective Peierls transition

Fundamental electronic principles underlying all transition metal compounds are the symmetry and filling of the d-electron orbitals and the influence of this filling on structural configurations and responses. Here we use a sensitive local structural technique, x-ray atomic pair distribution function analysis, to reveal the presence of fluctuating local-structural distortions at high temperature in one such compound, CuIr₂S₄. We show that this hitherto overlooked fluctuating symmetry-lowering is electronic in origin and will modify the energy-level spectrum and electronic and magnetic properties. The explanation is a local, fluctuating, orbital-degeneracy-lifted state. The natural extension of our result would be that this phenomenon is likely to be widespread amongst diverse classes of partially filled nominally degenerate d-electron systems, with potentially broad implications for our understanding of their properties.

A common feature of many transition metal materials is global symmetry breaking at low temperatures. Here the authors show that such materials are characterized by fluctuating symmetry-lowering distortions that exist pre-formed in higher temperature phases with greater average symmetry.

Light-Tunable 1T-TaS2 Charge-Density-Wave Oscillators

External stimuli-controlled phase transitions are essential for fundamental physics and design of functional devices. Charge density wave (CDW) is a metastable collective electronic phase featured by the periodic lattice distortion. Much attention has been attracted to study the external control of CDW phases. Although much work has been done in the electric-field-induced CDW transition, the study of the role of Joule heating in the phase transition is insufficient. Here, using the Raman spectroscopy, the electric-field-driven phase transition is in situ observed in the ultrathin 1T-TaS₂. By quantitative evaluation of the Joule heating effect in the electric-field-induced CDW transition, it is shown that Joule heating plays a secondary role in the nearly commensurate (NC) to incommensurate (IC) CDW transition, while it dominants the IC-NC CDW transition, providing a better understanding of the electric field-induced phase transition. More importantly, at room temperature, light illumination can modulate the CDW phase and thus tune the frequency of the ultrathin 1T-TaS₂ oscillators. This light tunability of the CDW phase transition is promising for multifunctional device applications.

Stabilization of three-dimensional charge order in YBa2Cu3O6+x via epitaxial growth

Incommensurate charge order (CO) has been identified as the leading competitor of high-temperature superconductivity in all major families of layered copper oxides, but the perplexing variety of CO states in different cuprates has confounded investigations of its impact on the transport and thermodynamic properties. The three-dimensional (3D) CO observed in YBa₂Cu₃O_6+x in high magnetic fields is of particular interest, because quantum transport measurements have revealed detailed information about the corresponding Fermi surface. Here we use resonant X-ray scattering to demonstrate 3D-CO in underdoped YBa₂Cu₃O_6+x films grown epitaxially on SrTiO₃ in the absence of magnetic fields. The resonance profiles indicate that Cu sites in the charge-reservoir layers participate in the CO state, and thus efficiently transmit CO correlations between adjacent CuO₂ bilayer units. The results offer fresh perspectives for experiments elucidating the influence of 3D-CO on the electronic properties of cuprates without the need to apply high magnetic fields.

In many cuprates the high temperature superconducting state competes with a charge ordered phase that has been difficult to investigate in detail. Here the authors show three-dimensional charge order can be stabilized in YBCO films and studied without using the high magnetic fields that are necessary in the bulk material.

Magnetic-field-tuned charge density wave in SmNiC2 and NdNiC2

We report magnetic field tuned competition between magnetic order and charge density wave (CDW) states in SmNiC₂ and NdNiC₂ polycrystals. The destruction of CDW can be observed not only in SmNiC₂ below ferromagnetic (FM) but also in NdNiC₂ below antiferromagnetic (AFM) transition temperature. Moreover, the CDW states near magnetic transition temperatures can be tuned by the magnetic field for both compounds. Magnetic-field induced FM state in NdNiC₂ is more effective in weakening the CDW than the AFM state at temperatures near Neel temperature T _N but both ordering states have the same effect on CDW below T _N. The interplay between magnetic and CDW states in SmNiC₂ and NdNiC₂ may be different, suggesting that these materials are good models to study correlations between magnetic and CDW wave order.

Anomalous dispersion X-ray diffraction study of Pb/Bi ordering/disordering states in PbTiO3-based perovskite oxides

Synchrotron radiation-based anomalous dispersion X-ray powder diffraction (ADSPD) was carried out to reveal the Pb/Bi ordering/disordering states in a series of PbTiO₃-based negative thermal expansion materials (1 - x)PbTiO₃ - xBiFeO₃ (x = 0.1, 0.3, 0.5) and (1 - x)PbTiO₃ - xBi(Zn_1/2Ti_1/2)O₃ (x = 0.1, 0.2, 0.3). It gives strong evidence of the disordered Pb/Bi distributions in these compositions, which is consistent with electron diffraction studies. Combined with binding energy calculation, we show that the disordered nature of Pb/Bi distributions is likely to be attributed to the similar electron configurations of Pb²⁺ and Bi³⁺ as well as their comparable coordinate environments in perovskite structures. The results of this study may be helpful to better understand the structure-property relationship in Pb/Bi-containing perovskites and are useful for further developing underlying physics in relevant materials.

Influence of the spatially inhomogeneous gap distribution on the quasiparticle current in c-axis junctions involving d-wave superconductors with charge density waves

The quasiparticle tunnel current J(V) between the superconducting ab-planes along the c-axis and the corresponding conductance [Formula: see text] were calculated for symmetric junctions composed of disordered d-wave layered superconductors partially gapped by charge density waves (CDWs). Here, V is the voltage. Both the checkerboard and unidirectional CDWs were considered. It was shown that the spatial spread of the CDW-pairing strength substantially smears the peculiarities of G(V) appropriate to uniform superconductors. The resulting curves G(V) become very similar to those observed for a number of cuprates in intrinsic junctions, e.g. mesas. In particular, the influence of CDWs may explain the peak-dip-hump structures frequently found for high-T c oxides.

Remarkable Stability of Charge Density Wave Order in La_{1.875}Ba_{0.125}CuO_{4}

The occurrence of charge-density-wave (CDW) order in underdoped cuprates is now well established, although the precise nature of the CDW and its relationship with superconductivity is not. Theoretical proposals include contrasting ideas such as that pairing may be driven by CDW fluctuations or that static CDWs may intertwine with a spatially modulated superconducting wave function. We test the dynamics of CDW order in La_{1.825}Ba_{0.125}CuO_{4} by using x-ray photon correlation spectroscopy at the CDW wave vector, detected resonantly at the Cu L_{3} edge. We find that the CDW domains are strikingly static, with no evidence of significant fluctuations up to 2 ¾  h. We discuss the implications of these results for some of the competing theories.