Charge order driven by Fermi-arc instability in Bi2Sr(2-x)La(x)CuO(6+δ)

Evolution from a charge-ordered insulator to a high-temperature superconductor in Bi2Sr2(Ca,Dy)Cu2O8+δ

How Cooper pairs form and condense has been the main challenge in the physics of copper-oxide high-temperature superconductors. Great efforts have been made in the 'underdoped' region of the phase diagram, through doping a Mott insulator or cooling a strange metal. However, there is still no consensus on how superconductivity emerges when electron-electron correlations dominate and the Fermi surface is missing. To address this issue, here we carry out high-resolution resonant inelastic X-ray scattering and scanning tunneling microscopy studies on prototype cuprates Bi2Sr2Ca0.6Dy0.4Cu2O8+δ near the onset of superconductivity, combining bulk and surface, momentum- and real-space information. We show that an incipient charge order exists in the antiferromagnetic regime down to 0.04 holes per CuO2 unit, entangled with a particle-hole asymmetric pseudogap. The charge order induces an intensity anomaly in the bond-buckling phonon branch, which exhibits an abrupt increase once the system enters the superconducting dome. Our results suggest that the Cooper pairs grow out of a charge-ordered insulating state, and then condense accompanied by an enhanced interplay between charge excitations and electron-phonon coupling.

Strain-induced long-range charge-density wave order in the optimally doped Bi2Sr2-xLaxCuO6 superconductor

The mechanism of high-temperature superconductivity in copper oxides (cuprate) remains elusive, with the pseudogap phase considered a potential factor. Recent attention has focused on a long-range symmetry-broken charge-density wave (CDW) order in the underdoped regime, induced by strong magnetic fields. Here by 63,65Cu-nuclear magnetic resonance, we report the discovery of a long-range CDW order in the optimally doped Bi2Sr2-xLaxCuO6 superconductor, induced by in-plane strain exceeding ∣ε∣ = 0.15 %, which deliberately breaks the crystal symmetry of the CuO2 plane. We find that compressive/tensile strains reduce superconductivity but enhance CDW, leaving superconductivity to coexist with CDW. The findings show that a long-range CDW order is an underlying hidden order in the pseudogap state, not limited to the underdoped regime, becoming apparent under strain. Our result sheds light on the intertwining of various orders in the cuprates.

Absence of 3a0 charge density wave order in the infinite-layer nickelate NdNiO2

A hallmark of many unconventional superconductors is the presence of many-body interactions that give rise to broken-symmetry states intertwined with superconductivity. Recent resonant soft X-ray scattering experiments report commensurate 3a0 charge density wave order in infinite-layer nickelates, which has important implications regarding the universal interplay between charge order and superconductivity in both cuprates and nickelates. Here we present X-ray scattering and spectroscopy measurements on a series of NdNiO2+x samples, which reveal that the signatures of charge density wave order are absent in fully reduced, single-phase NdNiO2. The 3a0 superlattice peak instead originates from a partially reduced impurity phase where excess apical oxygens form ordered rows with three-unit-cell periodicity. The absence of any observable charge density wave order in NdNiO2 highlights a crucial difference between the phase diagrams of cuprate and nickelate superconductors.

Correlation-Induced Symmetry-Broken States in Large-Angle Twisted Bilayer Graphene on MoS2

Strongly correlated states commonly emerge in twisted bilayer graphene (TBG) with "magic-angle" (1.1°), where the electron-electron (e-e) interaction U becomes prominent relative to the small bandwidth W of the nearly flat band. However, the stringent requirement of this magic angle makes the sample preparation and the further application facing great challenges. Here, using scanning tunneling microscopy (STM) and spectroscopy (STS), we demonstrate that the correlation-induced symmetry-broken states can also be achieved in a 3.45° TBG, via engineering this nonmagic-angle TBG into regimes of U/W > 1. We enhance the e-e interaction through controlling the microscopic dielectric environment by using a MoS2 substrate. Simultaneously, the width of the low-energy van Hove singularity (VHS) peak is reduced by enhancing the interlayer coupling via STM tip modulation. When partially filled, the VHS peak exhibits a giant splitting into two states flanked by the Fermi level and shows a symmetry-broken LDOS distribution with a stripy charge order, which confirms the existence of strong correlation effect in our 3.45° TBG. Our result demonstrates the feasibility of the study and application of the correlation physics in TBGs with a wider range of twist angle.

Disentangling Photodoping, Photoconductivity, and Photosuperconductivity in the Cuprates

The normal-state conductivity and superconducting critical temperature of oxygen-deficient YBa_{2}Cu_{3}O_{7-δ} can be persistently enhanced by illumination. Strongly debated for years, the origin of those effects-termed persistent photoconductivity and photosuperconductivity (PPS)-has remained an unsolved critical problem, whose comprehension may provide key insights to harness the origin of high-temperature superconductivity itself. Here, we make essential steps toward understanding PPS. While the models proposed so far assume that it is caused by a carrier-density increase (photodoping) observed concomitantly, our experiments contradict such conventional belief: we demonstrate that it is instead linked to a photo-induced decrease of the electronic scattering rate. Furthermore, we find that the latter effect and photodoping are completely disconnected and originate from different microscopic mechanisms, since they present different wavelength and oxygen-content dependences as well as strikingly different relaxation dynamics. Besides helping disentangle photodoping, persistent photoconductivity, and PPS, our results provide new evidence for the intimate relation between critical temperature and scattering rate, a key ingredient in modern theories on high-temperature superconductivity.

Universal Stripe Symmetry of Short-Range Charge Density Waves in Cuprate Superconductors

The omnipresence of charge density waves (CDWs) across almost all cuprate families underpins a common organizing principle. However, a longstanding debate of whether its spatial symmetry is stripe or checkerboard remains unresolved. While CDWs in lanthanum- and yttrium-based cuprates possess a stripe symmetry, distinguishing these two scenarios is challenging for the short-range CDW in bismuth-based cuprates. Here, high-resolution resonant inelastic x-ray scattering is employed to uncover the spatial symmetry of the CDW in Bi2 Sr2 - x Lax CuO6 + δ . Across a wide range of doping and temperature, anisotropic CDW peaks with elliptical shapes are found in reciprocal space. Based on Fourier transform analysis of real-space models, the results are interpreted as evidence of unidirectional charge stripes, hosted by mutually 90°-rotated anisotropic domains. This work paves the way for a unified symmetry and microscopic description of CDW order in cuprates.

Signature of quantum criticality in cuprates by charge density fluctuations

The universality of the strange metal phase in many quantum materials is often attributed to the presence of a quantum critical point (QCP), a zero-temperature phase transition ruled by quantum fluctuations. In cuprates, where superconductivity hinders direct QCP observation, indirect evidence comes from the identification of fluctuations compatible with the strange metal phase. Here we show that the recently discovered charge density fluctuations (CDF) possess the right properties to be associated to a quantum phase transition. Using resonant x-ray scattering, we studied the CDF in two families of cuprate superconductors across a wide doping range (up to p = 0.22). At p* ≈ 0.19, the putative QCP, the CDF intensity peaks, and the characteristic energy Δ is minimum, marking a wedge-shaped region in the phase diagram indicative of a quantum critical behavior, albeit with anomalies. These findings strengthen the role of charge order in explaining strange metal phenomenology and provide insights into high-temperature superconductivity.

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.

Unveiling phase diagram of the lightly doped high-Tc cuprate superconductors with disorder removed

The currently established electronic phase diagram of cuprates is based on a study of single- and double-layered compounds. These CuO₂ planes, however, are directly contacted with dopant layers, thus inevitably disordered with an inhomogeneous electronic state. Here, we solve this issue by investigating a 6-layered Ba₂Ca₅Cu₆O₁₂(F,O)₂ with inner CuO₂ layers, which are clean with the extremely low disorder, by angle-resolved photoemission spectroscopy (ARPES) and quantum oscillation measurements. We find a tiny Fermi pocket with a doping level less than 1% to exhibit well-defined quasiparticle peaks which surprisingly lack the polaronic feature. This provides the first evidence that the slightest amount of carriers is enough to turn a Mott insulating state into a metallic state with long-lived quasiparticles. By tuning hole carriers, we also find an unexpected phase transition from the superconducting to metallic states at 4%. Our results are distinct from the nodal liquid state with polaronic features proposed as an anomaly of the heavily underdoped cuprates.

Robust charge-density-wave correlations in the electron-doped single-band Hubbard model

There is growing evidence that the hole-doped single-band Hubbard and t - J models do not have a superconducting ground state reflective of the high-temperature cuprate superconductors but instead have striped spin- and charge-ordered ground states. Nevertheless, it is proposed that these models may still provide an effective low-energy model for electron-doped materials. Here we study the finite temperature spin and charge correlations in the electron-doped Hubbard model using quantum Monte Carlo dynamical cluster approximation calculations and contrast their behavior with those found on the hole-doped side of the phase diagram. We find evidence for a charge modulation with both checkerboard and unidirectional components decoupled from any spin-density modulations. These correlations are inconsistent with a weak-coupling description based on Fermi surface nesting, and their doping dependence agrees qualitatively with resonant inelastic x-ray scattering measurements. Our results provide evidence that the single-band Hubbard model describes the electron-doped cuprates.

Critical nematic correlations throughout the superconducting doping range in Bi2-zPbzSr2-yLayCuO6+x

Charge modulations have been widely observed in cuprates, suggesting their centrality for understanding the high-T_c superconductivity in these materials. However, the dimensionality of these modulations remains controversial, including whether their wavevector is unidirectional or bidirectional, and also whether they extend seamlessly from the surface of the material into the bulk. Material disorder presents severe challenges to understanding the charge modulations through bulk scattering techniques. We use a local technique, scanning tunneling microscopy, to image the static charge modulations on Bi_2-zPb_zSr_2-yLa_yCuO_6+x. The ratio of the phase correlation length ξ_CDW to the orientation correlation length ξ_orient points to unidirectional charge modulations. By computing new critical exponents at free surfaces including that of the pair connectivity correlation function, we show that these locally 1D charge modulations are actually a bulk effect resulting from classical 3D criticality of the random field Ising model throughout the entire superconducting doping range.

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.

Observations of Nematicity, Dopants, and Zero-Bias Conductance Peaks for the Ca0.9La0.1FeAs2 Superconductor

Ca_1-xLa_xFeAs₂ (CLFA112) belongs to a new family of Fe-based superconductors (FeSCs) and has a unique crystal structure featuring an arsenic zigzag chain layer, which has been proposed to be a possible two-dimensional topological insulator. This suggests that CLFA112 is a potential topological superconductor-a platform to realize Majorana fermions. Up to now, even a clear superconducting (SC) gap in CLFA112 has never been observed, and the SC properties of CLFA112 remain largely elusive. In this letter, we report the results of an atomic-scale investigation of the electronic structure of CLFA112 crystals using low-temperature scanning tunneling microscopy (STM). We revealed four different types of surfaces exhibiting distinct electronic properties, with all surfaces displaying dominating 2 × 1 surface reconstructions. On a Ca/La layer on top of an FeAs layer, a clear SC gap of ~12 mV was observed only at the crevices (vacancies) where the FeAs layer can be directly accessed. Remarkably, the FeAs termination layer displayed a dispersing nematic modulation both in real and q space. We also present peculiar zero-bias conductance peaks for the very As chain layer that is believed to exhibit a topological edge state as well as the influence of La dopants on the As chain layer.

Fate of charge order in overdoped La-based cuprates

In high-temperature cuprate superconductors, stripe order refers broadly to a coupled spin and charge modulation with a commensuration of eight and four lattice units, respectively. How this stripe order evolves across optimal doping remains a controversial question. Here we present a systematic resonant inelastic x-ray scattering study of weak charge correlations in La2-xSrxCuO4 and La1.8-xEu0.2SrxCuO4. Ultra high energy resolution experiments demonstrate the importance of the separation of inelastic and elastic scattering processes. Long-range temperature-dependent stripe order is only found below optimal doping. At higher doping, short-range temperature-independent correlations are present up to the highest doping measured. This transformation is distinct from and preempts the pseudogap critical doping. We argue that the doping and temperature-independent short-range correlations originate from unresolved electron-phonon coupling that broadly peaks at the stripe ordering vector. In La2-xSrxCuO4, long-range static stripe order vanishes around optimal doping and we discuss both quantum critical and crossover scenarios.

Stabilization of three-dimensional charge order through interplanar orbital hybridization in PrxY1-xBa2Cu3O6+δ

The shape of 3d-orbitals often governs the electronic and magnetic properties of correlated transition metal oxides. In the superconducting cuprates, the planar confinement of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d}_{{x}^{2}-{y}^{2}}$$\end{document}dx2−y2 orbital dictates the two-dimensional nature of the unconventional superconductivity and a competing charge order. Achieving orbital-specific control of the electronic structure to allow coupling pathways across adjacent planes would enable direct assessment of the role of dimensionality in the intertwined orders. Using Cu L₃ and Pr M₅ resonant x-ray scattering and first-principles calculations, we report a highly correlated three-dimensional charge order in Pr-substituted YBa₂Cu₃O₇, where the Pr f-electrons create a direct orbital bridge between CuO₂ planes. With this we demonstrate that interplanar orbital engineering can be used to surgically control electronic phases in correlated oxides and other layered materials.

External perturbations can induce 3D charge order in cuprates, but the 3D correlation length is limited and the mechanism is not well understood. Ruiz et al. show that Pr substitution in YBa₂Cu₃O₇ enhances interplanar orbital coupling and stabilizes coherent 3D charge order that coexists with superconductivity.

Pseudogap in underdoped cuprate seen in longitudinal magnetoresistance

We report the results of in-plane magnetotransport study of slightly underdoped cuprate La_1.85Sr_0.15CuO₄(LSCO15) with Ni impurity. Increasing Ni contentycauses a sharp drop in longitudinal magnetoresistance (LMR) in LSCO15 to broaden and move towards higher temperatures. TemperatureTmLMR(y)of this local maximum in LMR coincides with temperatureTdev(y), below which ideal resistivity from the parallel-resistor model deviates from itsT²-dependence. A direct comparison with the hole doping evolution of pseudogap (PG) in La2-xSr_xCuO₄(LSCO), possible through the mobile-carrier concentration extracted from the thermopower measurements, allows to equate both characteristic temperaturesTmLMR≅Tdevwith PG opening temperatureT∗. The rate of PG closing by magnetic field parallel to the CuO₂plane, in measurements up to 9 T, is consistent with spin-paramagnetic effect in this configuration and yields PG closing fieldB_pcclose to the second critical fieldBc2predicted for superconducting gap with the help of Werthamer-Helfand-Hohenberg theory. The field anisotropy ofB_pcsuggests that orbital degrees of freedom also play a role in PG formation.

Recent Development of Ultrafast Optical Characterizations for Quantum Materials

The advent of intense ultrashort optical pulses spanning a frequency range from terahertz to the visible has opened a new era in the experimental investigation and manipulation of quantum materials. The generation of strong optical field in an ultrashort time scale enables the steering of quantum materials nonadiabatically, inducing novel phenomenon or creating new phases which may not have an equilibrium counterpart. Ultrafast time-resolved optical techniques have provided rich information and played an important role in characterization of the nonequilibrium and nonlinear properties of solid systems. Here, some of the recent progress of ultrafast optical techniques and their applications to the detection and manipulation of physical properties in selected quantum materials are reviewed. Specifically, the new development in the detection of the Higgs mode and photoinduced nonequilibrium response in the study of superconductors by time-resolved terahertz spectroscopy are discussed.

Tracking the nematicity in cuprate superconductors: a resistivity study under uniaxial pressure

Overshadowing the superconducting dome in hole-doped cuprates, the pseudogap state is still one of the mysteries that no consensus can be achieved. It has been suggested that the rotational symmetry is broken in this state and may result in a nematic phase transition, whose temperature seems to coincide with the onset temperature of the pseudogap stateT∗around optimal doping level, raising the question whether the pseudogap results from the establishment of the nematic order. Here we report results of resistivity measurements under uniaxial pressure on several hole-doped cuprates, where the normalized slope of the elastoresistivityζcan be obtained as illustrated in iron-based superconductors. The temperature dependence ofζalong particular lattice axis exhibits kink feature atT_kand shows Curie-Weiss-like behavior above it, which may suggest a spontaneous nematic transition. WhileT_kseems to be the same asT∗around the optimal doping and in the overdoped region, they become very different in underdoped La2-xSr_xCuO₄. Our results suggest that the nematic order, if indeed existing, is an electronic phase within the pseudogap state.

Generic character of charge and spin density waves in superconducting cuprates

The essential physics of the cuprate high-temperature superconductors have been a central focus of condensed-matter physics for more than three decades. Although initially controversial, it is now clear that a ubiquitous tendency toward charge-density-wave (CDW) order is intertwined with the superconductivity. However, this manifests differently in distinct cuprates. On the basis of extensive X-ray and neutron scattering studies of the temperature and doping dependence of the CDW and spin-density-wave (SDW) correlations in one representative cuprate and a comparison with existing studies on other cuprates, we show that there plausibly is a single, preferred CDW order at the microscale, whose manifestation at low temperatures is modified in predictable ways by material-specific details, including its interaction with SDW order.

Charge density waves (CDWs) have been observed in nearly all families of copper-oxide superconductors. But the behavior of these phases across different families has been perplexing. In La-based cuprates, the CDW wavevector is an increasing function of doping, exhibiting the so-called Yamada behavior, while in Y- and Bi-based materials the behavior is the opposite. Here, we report a combined resonant soft X-ray scattering (RSXS) and neutron scattering study of charge and spin density waves in isotopically enriched La1.8−xEu0.2Sr_xCuO₄ over a range of doping 0.07≤x≤0.20. We find that the CDW amplitude is temperature independent and develops well above experimentally accessible temperatures. Further, the CDW wavevector shows a nonmonotonic temperature dependence, exhibiting Yamada behavior at low temperature with a sudden change occurring near the spin ordering temperature. We describe these observations using a Landau–Ginzburg theory for an incommensurate CDW in a metallic system with a finite charge compressibility and spin-CDW coupling. Extrapolating to high temperature, where the CDW amplitude is small and spin order is absent, our analysis predicts a decreasing wavevector with doping, similar to Y and Bi cuprates. Our study suggests that CDW order in all families of cuprates forms by a common mechanism.

The Strange-Metal Behavior of Cuprates

Recent resonant X-ray scattering experiments on cuprates allowed to identify a new kind of collective excitations, known as charge density fluctuations, which have finite characteristic wave vector, short correlation length and small characteristic energy. It was then shown that these fluctuations provide a microscopic scattering mechanism that accounts for the anomalous transport properties of cuprates in the so-called strange-metal phase and are a source of anomalies in the specific heat. In this work, we retrace the main steps that led us to attributing a central role to charge density fluctuations in the strange-metal phase of cuprates, discuss the state of the art on the issue and provide an in-depth analysis of the contribution of charge density fluctuations to the specific heat.

Characterization of photoinduced normal state through charge density wave in superconducting YBa2Cu3O6.67

The normal state of high-T_c cuprates has been considered one of the essential topics in high-temperature superconductivity research. However, compared to the high magnetic field study of it, understanding a photoinduced normal state remains elusive. Here, we explore a photoinduced normal state of YBa₂Cu₃O_6.67 through a charge density wave (CDW) with time-resolved resonant soft x-ray scattering, as well as a high magnetic field x-ray scattering. In the nonequilibrium state where people predict a quenched superconducting state based on the previous optical spectroscopies, we experimentally observed a similar analogy to the competition between superconductivity and CDW shown in the equilibrium state. We further observe that the broken pairing states in the superconducting CuO₂ plane via the optical pump lead to nucleation of three-dimensional CDW precursor correlation. Ultimately, these findings provide a critical clue that the characteristics of the photoinduced normal state show a solid resemblance to those under magnetic fields in equilibrium conditions.

Charge density waves and Fermi surface reconstruction in the clean overdoped cuprate superconductor Tl2Ba2CuO6+δ

Hall effect and quantum oscillation measurements on high temperature cuprate superconductors show that underdoped compositions have small Fermi surface pockets whereas when heavily overdoped, a single much larger pocket is found. The origin of this change in electronic structure has been unclear, but may be related to the high temperature superconductivity. Here we show that the clean overdoped single-layer cuprate Tl₂Ba₂CuO_6+δ (Tl2201) displays CDW order with a remarkably long correlation length ξ ≈ 200 Å which disappears above a hole doping of p_CDW ≈ 0.265. We show that the evolution of the electronic properties of Tl2201 as the doping is lowered may be explained by a Fermi surface reconstruction which accompanies the emergence of the CDW below p_CDW. Our results demonstrate importance of CDW correlations in understanding the electronic properties of overdoped cuprates.

The origin of the Fermi surface reconstruction that occurs in cuprate superconductors as hole doping increases remains unclear. Here, the authors observe long range charge density wave (CDW) order in the overdoped single-layer cuprate Tl₂Ba₂CuO_6+δ, which then disappears above a hole concentration 0.265, suggesting a correlation between Fermi surface reconstruction and the emergence of the CDW.

Simultaneous Control of Bandfilling and Bandwidth in Electric Double-Layer Transistor Based on Organic Mott Insulator κ-(BEDT-TTF)2Cu[N(CN)2]Cl

The physics of quantum many-body systems have been studied using bulk correlated materials, and recently, moiré superlattices formed by atomic bilayers have appeared as a novel platform in which the carrier concentration and the band structures are highly tunable. In this brief review, we introduce an intermediate platform between those systems, namely, a band-filling- and bandwidth-tunable electric double-layer transistor based on a real organic Mott insulator κ-(BEDT-TTF)2Cu[N(CN)2]Cl. In the proximity of the bandwidth-control Mott transition at half filling, both electron and hole doping induced superconductivity (with almost identical transition temperatures) in the same sample. The normal state under electric double-layer doping exhibited non-Fermi liquid behaviors as in many correlated materials. The doping levels for the superconductivity and the non-Fermi liquid behaviors were highly doping-asymmetric. Model calculations based on the anisotropic triangular lattice explained many phenomena and the doping asymmetry, implying the importance of the noninteracting band structure (particularly the flat part of the band).

Strange bedfellows inside a superconductor

[Figure: see text].

Atomically-resolved interlayer charge ordering and its interplay with superconductivity in YBa2Cu3O6.81

High-temperature superconductive (SC) cuprates exhibit not only a SC phase, but also competing orders, suppressing superconductivity. Charge order (CO) has been recognized as an important competing order, but its microscopic spatial interplay with SC phase as well as the interlayer coupling in CO and SC phases remain elusive, despite being essential for understanding the physical mechanisms of competing orders and hence superconductivity. Here we report the achievement of direct real-space imaging with atomic-scale resolution of cryogenically cleaved YBa₂Cu₃O_6.81 using cross-sectional scanning tunneling microscopy/spectroscopy. CO nanodomains are found embedded in the SC phase with a proximity-like boundary region characterized by mutual suppression of CO and superconductivity. Furthermore, SC coherence as well as CO occur on both CuO chain and plane layers, revealing carrier transport and density of states mixing between layers. The CO antiphase correlation along the c direction suggests a dominance of Coulomb repulsion over Josephson tunneling between adjacent layers.

Charge ordering and superconductivity are known to compete in layered cuprates; however, precise real-space characterization of their interplay has been lacking. Here, the authors address this using atomically-resolved cross-sectional scanning tunnelling microscopy and spectroscopy on cryogenically cleaved YBa₂Cu₃O₆.₈₁.

Locally commensurate charge-density wave with three-unit-cell periodicity in YBa2Cu3Oy

In order to identify the mechanism responsible for the formation of charge-density waves (CDW) in cuprate superconductors, it is important to understand which aspects of the CDW's microscopic structure are generic and which are material-dependent. Here, we show that, at the local scale probed by NMR, long-range CDW order in YBa2Cu3Oy is unidirectional with a commensurate period of three unit cells (λ = 3b), implying that the incommensurability found in X-ray scattering is ensured by phase slips (discommensurations). Furthermore, NMR spectra reveal a predominant oxygen character of the CDW with an out-of-phase relationship between certain lattice sites but no specific signature of a secondary CDW with λ = 6b associated with a putative pair-density wave. These results shed light on universal aspects of the cuprate CDW. In particular, its spatial profile appears to generically result from the interplay between an incommensurate tendency at long length scales, possibly related to properties of the Fermi surface, and local commensuration effects, due to electron-electron interactions or lock-in to the lattice.

Quasi-One-Dimensional Fermi Surface Nesting and Hidden Nesting Enable Multiple Kohn Anomalies in α-Uranium

The topology of the Fermi surface controls the electronic response of a metal, including charge density wave (CDW) formation. A topology conducive for Fermi surface nesting (FSN) allows the electronic susceptibility χ_{0} to diverge and induce a CDW at wave vector q_{CDW}. Kohn extended the implications of FSN to show that the imaginary part of the lattice dynamical susceptibility χ_{L}^{''} also responds anomalously for all phonon branches at q_{CDW}-a phenomenon referred to as the Kohn anomaly. However, materials exhibiting multiple Kohn anomalies remain rare. Using first-principles simulations of χ_{0} and χ_{L}^{''}, and previous scattering measurements [Crummett et al., Phys. Rev. B 19, 6028 234 (1979)PRBMDO0163-1829], we show that α-uranium harbors multiple Kohn anomalies enabled by the combined effect of FSN and "hidden" nesting, i.e., nesting of electronic states above and below the Fermi surface. FSN and hidden nesting lead to a ridgelike feature in the real part of χ_{0}, allowing interatomic forces to modulate strongly and multiple Kohn anomalies to emerge. These results emphasize the importance of hidden nesting in controlling χ_{0} and χ_{L}^{''} to exploit electronic and lattice states and enable engineering of advanced materials, including topological Weyl semimetals and superconductors.

Fluctuation induced conductivity and pseudogap state studies of Bi1.6Pb0.4Sr2Ca2Cu3O10+δ superconductor added with ZnO nanoparticles

The major limitations of the Bi_1.6Pb_0.4Sr₂Ca₂Cu₃O_10+δ superconductor are weak flux pinning capability and weak inter-grains coupling that lead to a low critical current density and low critical magnetic field which impedes the suppleness of this material towards practical applications. The addition of nanoscales impurities can create artificial pining centers that may improve flux pinning capability and intergranular coupling. In this work, the influences of ZnO nanoparticles on the superconducting parameters and pseudogap properties of the Bi_1.6Pb_0.4Sr₂Ca₂Cu₃O_10+δ superconductor are investigated using fluctuation induced conductivity analyses. Results demonstrate that the ZnO nanoparticles addition improves the formation of the Bi_1.6Pb_0.4Sr₂Ca₂Cu₃O_10+δ phase significantly. Various superconducting parameters include coherence length along c-axis (ξ_c(0)), penetration depth (λ_pd(0)), Fermi velocity (v_F), Fermi energy (E_F), lower and upper critical magnetic fields (B_c1(0) and B_c2(0) respectively) and critical current density (J_c(0)), are estimated for samples with different amounts of ZnO nanoparticles. It is found that the values of the B_c1(0), B_c2(0), and J_c(0) are improved significantly in the 0.2 wt% ZnO added sample in comparison to the ZnO-free sample. The magnitude and temperature dependence of the pseudogap Δ*(T) is calculated using the local pairs model. The obtained values of T_pair, the temperature at which local pairs are transformed from strongly coupled bosons into the fluctuating Cooper pairs, increases as the added ZnO nanoparticles concentration enhances up to 0.2 wt%. Also, the estimated values for the superconducting gap at T = 0 K (Δ(0)) are decreased from about 26 meV in ZnO-free sample to about 22 meV in 0.2 wt% ZnO added sample and then increases for higher values of additive.

Anomalous vortex liquid in charge-ordered cuprate superconductors

The magnetic-field scale at which superconducting vortices persist in underdoped cuprate superconductors has remained a controversial subject. Here we present an electrical transport study on three distinctly different cuprate families, at temperatures down to 0.32 K and magnetic fields up to 45 T. We reveal the presence of an anomalous vortex liquid state with a highly nonohmic resistivity in all three materials, irrespective of the level of disorder or structural details. The doping and field regime over which this anomalous vortex state persists suggests its occurrence is tied to the presence of long-range charge order under high magnetic field. Our results demonstrate that the intricate interplay between charge order and superconductivity can lead to an exotic vortex state.

The interplay between charge order and d-wave superconductivity in high-Tc cuprates remains an open question. While mounting evidence from spectroscopic probes indicates that charge order competes with superconductivity, to date little is known about the impact of charge order on charge transport in the mixed state, when vortices are present. Here we study the low-temperature electrical resistivity of three distinctly different cuprate families under intense magnetic fields, over a broad range of hole doping and current excitations. We find that the electronic transport in the doping regime where long-range charge order is known to be present is characterized by a nonohmic resistivity, the identifying feature of an anomalous vortex liquid. The field and temperature range in which this nonohmic behavior occurs indicates that the presence of long-range charge order is closely related to the emergence of this anomalous vortex liquid, near a vortex solid boundary that is defined by the excitation current in the T→ 0 limit. Our findings further suggest that this anomalous vortex liquid, a manifestation of fragile superconductivity with a suppressed critical current density, is ubiquitous in the high-field state of charge-ordered cuprates.

Dynamic electron correlations with charge order wavelength along all directions in the copper oxide plane

In strongly correlated systems the strength of Coulomb interactions between electrons, relative to their kinetic energy, plays a central role in determining their emergent quantum mechanical phases. We perform resonant x-ray scattering on Bi₂Sr₂CaCu₂O_8+δ, a prototypical cuprate superconductor, to probe electronic correlations within the CuO₂ plane. We discover a dynamic quasi-circular pattern in the x-y scattering plane with a radius that matches the wave vector magnitude of the well-known static charge order. Along with doping- and temperature-dependent measurements, our experiments reveal a picture of charge order competing with superconductivity where short-range domains along x and y can dynamically rotate into any other in-plane direction. This quasi-circular spectrum, a hallmark of Brazovskii-type fluctuations, has immediate consequences to our understanding of rotational and translational symmetry breaking in the cuprates. We discuss how the combination of short- and long-range Coulomb interactions results in an effective non-monotonic potential that may determine the quasi-circular pattern.

Knowledge of effective Coulomb interactions is central to understand emergent quantum phases in strongly correlated systems. Here, Boschini et al. report a dynamic quasi-circular spectrum of charge density wave fluctuations in the CuO₂ plane of Bi₂Sr₂CaCu₂O_8+δ, shedding a light on understanding how Coulomb interactions can lead to rotational and translational symmetry breaking in the cuprates.

Charge Density Waves in YBa_{2}Cu_{3}O_{6.67} Probed by Resonant X-Ray Scattering under Uniaxial Compression

We report a comprehensive Cu L_{3}-edge resonant x-ray scattering (RXS) study of two- and three-dimensional (2D and 3D) incommensurate charge correlations in single crystals of the underdoped high-temperature superconductor YBa_{2}Cu_{3}O_{6.67} under uniaxial compression up to 1% along the two inequivalent Cu─O─Cu bond directions (a and b) in the CuO_{2} planes. We confirm the strong in-plane anisotropy of the 2D charge correlations and observe their symmetric response to pressure: pressure along a enhances correlations along b, and vice versa. Our results imply that the underlying order parameter is uniaxial. In contrast, 3D long-range charge order is only observed along b in response to compression along a. Spectroscopic RXS measurements show that the 3D charge order resides exclusively in the CuO_{2} planes and may thus be generic to the cuprates. We discuss implications of these results for models of electronic nematicity and for the interplay between charge order and superconductivity.

Revival of Charge Density Waves and Charge Density Fluctuations in Cuprate High-Temperature Superconductors

I present here a short memory of my scientific contacts with K.A. Müller starting from the Interlaken Conference (1988), Erice (1992 and 1993), and Cottbus (1994) on the initial studies on phase separation (PS) and charge inhomogeneity in cuprates carried out against the view of the majority of the scientific community at that time. Going over the years and passing through the charge density wave (CDW) instability of the correlated Fermi liquid (FL) and to the consequences of charge density fluctuations (CDFs), I end with a presentation of my current research activity on CDWs and the related two-dimensional charge density fluctuations (2D-CDFs). A scenario follows of the physics of cuprates, which includes the solution of the decades-long problem of the strange metal (SM) state.

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.

Phase Separation and Pairing Fluctuations in Oxide Materials

The microscopic mechanism of charge instabilities and the formation of inhomogeneous states in systems with strong electron correlations is investigated. We demonstrate that within a strong coupling expansion the single-band Hubbard model shows an instability towards phase separation and extend the approach also for an analysis of phase separation in the Hubbard-Kanamori hamiltonian as a prototypical multiband model. We study the pairing fluctuations on top of an inhomogeneous stripe state where superconducting correlations in the extended s-wave and d-wave channels correspond to (anti)bound states in the two-particle spectra. Whereas extended s-wave fluctuations are relevant on the scale of the local interaction parameter U, we find that d-wave fluctuations are pronounced in the energy range of the active subband which crosses the Fermi level. As a result, low energy spin and charge fluctuations can transfer the d-wave correlations from the bound states to the low energy quasiparticle bands. Our investigations therefore help to understand the coexistence of stripe correlations and d-wave superconductivity in cuprates.

Enhanced Electron-Phonon Coupling for Charge-Density-Wave Formation in La_{1.8-x}Eu_{0.2}Sr_{x}CuO_{4+δ}

Charge density wave (CDW) correlations are prevalent in all copper-oxide superconductors. While CDWs in conventional metals are driven by coupling between lattice vibrations and electrons, the role of the electron-phonon coupling (EPC) in cuprate CDWs is strongly debated. Using Cu L_{3} edge resonant inelastic x-ray scattering, we study the CDW and Cu-O bond-stretching phonons in the stripe-ordered cuprate La_{1.8-x}Eu_{0.2}Sr_{x}CuO_{4+δ}. We investigate the interplay between charge order and EPC as a function of doping and temperature and find that the EPC is enhanced in a narrow momentum region around the CDW ordering vector. By detuning the incident photon energy from the absorption resonance, we extract an EPC matrix element at the CDW ordering vector of M≃0.36  eV, which decreases to M≃0.30  eV at high temperature in the absence of the CDW. Our results suggest a feedback mechanism in which the CDW enhances the EPC which, in turn, further stabilizes the CDW.

Observation of small Fermi pockets protected by clean CuO2 sheets of a high-T c superconductor

In cuprate superconductors with high critical transition temperature (T _c), light hole-doping to the parent compound, which is an antiferromagnetic Mott insulator, has been predicted to lead to the formation of small Fermi pockets. These pockets, however, have not been observed. Here, we investigate the electronic structure of the five-layered Ba₂Ca₄Cu₅O₁₀(F,O)₂, which has inner copper oxide (CuO₂) planes with extremely low disorder, and find small Fermi pockets centered at (π/2, π/2) of the Brillouin zone by angle-resolved photoemission spectroscopy and quantum oscillation measurements. The d-wave superconducting gap opens along the pocket, revealing the coexistence between superconductivity and antiferromagnetic ordering in the same CuO₂ sheet. These data further indicate that superconductivity can occur without contribution from the antinodal region around (π, 0), which is shared by other competing excitations.

A framework for the description of charge order, pseudo and superconducting gap, critical temperature and pairing interaction in cuprate superconductors

A unified phenomenological description framework is proposed for the evaluation of some of the most important observables of the cuprate superconductors: the pseudogap (PG) ΔPG, the local superconducting amplitudes ΔSC(ri), the critical temperatureTcand charge ordering (CO) parameters. Recent detailed measurements of CO structures and CO wavelengthsλCOare faithfully reproduced by solutions of a Cahn-Hilliard differential equation with a free energy potentialVGLthat produces alternating small charge modulations. The charge oscillations induce atomic fluctuations that mediate the SC pair interaction proportional to theVGLamplitude. The local SC amplitude and phaseθiare connected by Josephson couplingEJ(rij) and the SC long-range order transition occurs whenEJ∼kBTc. The calculated results of the wavelengthλCO, ΔPG,ΔSCandTccalculations are in good agreement with a variety of experiments.

Atomic-scale electronic structure of the cuprate pair density wave state coexisting with superconductivity

By making a variety of quantitative comparisons between electronic visualization experiments and a theory describing coexisting pair density wave and superconductive states in cuprates, we find striking correspondence throughout. Our model can thus explain the microscopic origins of many key atomic-scale phenomena of the cuprate broken-symmetry state. These observations are consistent with the possibility that a short-range pair density wave (PDW) state coexists with superconductivity below a critical hole density in Bi₂Sr₂CaCu₂O₈, that the charge density wave modulations in cuprates are a consequence of the PDW state, that the cuprate pseudogap is the antinodal gap of the PDW, and that the critical point in the cuprate phase diagram occurs due to disappearance of the PDW.

The defining characteristic of hole-doped cuprates is d-wave high temperature superconductivity. However, intense theoretical interest is now focused on whether a pair density wave state (PDW) could coexist with cuprate superconductivity [D. F. Agterberg et al., Annu. Rev. Condens. Matter Phys. 11, 231 (2020)]. Here, we use a strong-coupling mean-field theory of cuprates, to model the atomic-scale electronic structure of an eight-unit-cell periodic, d-symmetry form factor, pair density wave (PDW) state coexisting with d-wave superconductivity (DSC). From this PDW + DSC model, the atomically resolved density of Bogoliubov quasiparticle states Nr,E is predicted at the terminal BiO surface of Bi₂Sr₂CaCu₂O₈ and compared with high-precision electronic visualization experiments using spectroscopic imaging scanning tunneling microscopy (STM). The PDW + DSC model predictions include the intraunit-cell structure and periodic modulations of Nr,E, the modulations of the coherence peak energy Δpr, and the characteristics of Bogoliubov quasiparticle interference in scattering-wavevector space q-space. Consistency between all these predictions and the corresponding experiments indicates that lightly hole-doped Bi₂Sr₂CaCu₂O₈ does contain a PDW + DSC state. Moreover, in the model the PDW + DSC state becomes unstable to a pure DSC state at a critical hole density p*, with empirically equivalent phenomena occurring in the experiments. All these results are consistent with a picture in which the cuprate translational symmetry-breaking state is a PDW, the observed charge modulations are its consequence, the antinodal pseudogap is that of the PDW state, and the cuprate critical point at p* ≈ 19% occurs due to disappearance of this PDW.

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.

Calculations of quantum oscillations in cuprate superconductors considering the pseudogap

The observations of quantum oscillations frequencies in overdoped cuprates were in agreement with a charge density contained in a cylindrical Fermi surface but the measured frequencies of underdoped compounds were much smaller than expected. This was attributed to a topological transition into small pockets of Fermi surface associated with the existence of charge density waves. On the other hand, spectroscopic measurements suggested that the large two-dimensional Fermi surface changes continuously into a set of four disconnected arcs. Here we take into account the effect of the pseudogap that limits the availablek-space area where the Landau levels are developed on the Luttinger theorem and obtain the correct total carrier densities. The calculations show how the disconnected arcs evolve into a closed Fermi surface reconciling the experiments.

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.

Devil's staircase transition of the electronic structures in CeSb

Solids with competing interactions often undergo complex phase transitions with a variety of long-periodic modulations. Among such transition, devil's staircase is the most complex phenomenon, and for it, CeSb is the most famous material, where a number of the distinct phases with long-periodic magnetostructures sequentially appear below the Néel temperature. An evolution of the low-energy electronic structure going through the devil's staircase is of special interest, which has, however, been elusive so far despite 40 years of intense research. Here, we use bulk-sensitive angle-resolved photoemission spectroscopy and reveal the devil's staircase transition of the electronic structures. The magnetic reconstruction dramatically alters the band dispersions at each transition. Moreover, we find that the well-defined band picture largely collapses around the Fermi energy under the long-periodic modulation of the transitional phase, while it recovers at the transition into the lowest-temperature ground state. Our data provide the first direct evidence for a significant reorganization of the electronic structures and spectral functions occurring during the devil's staircase.

Strongly Correlated Charge Density Wave in La_{2-x}Sr_{x}CuO_{4} Evidenced by Doping-Dependent Phonon Anomaly

The discovery of charge-density-wave-related effects in the resonant inelastic x-ray scattering spectra of cuprates holds the tantalizing promise of clarifying the interactions that stabilize the electronic order. Here, we report a comprehensive resonant inelastic x-ray scattering study of La_{2-x}Sr_{x}CuO_{4} finding that charge-density wave effects persist up to a remarkably high doping level of x=0.21 before disappearing at x=0.25. The inelastic excitation spectra remain essentially unchanged with doping despite crossing a topological transition in the Fermi surface. This indicates that the spectra contain little or no direct coupling to electronic excitations near the Fermi surface, rather they are dominated by the resonant cross section for phonons and charge-density-wave-induced phonon softening. We interpret our results in terms of a charge-density wave that is generated by strong correlations and a phonon response that is driven by the charge-density-wave-induced modification of the lattice.

Dynamical charge density fluctuations pervading the phase diagram of a Cu-based high-T c superconductor

Charge density modulations have been observed in all families of high-critical temperature (T _c) superconducting cuprates. Although they are consistently found in the underdoped region of the phase diagram and at relatively low temperatures, it is still unclear to what extent they influence the unusual properties of these systems. Using resonant x-ray scattering, we carefully determined the temperature dependence of charge density modulations in YBa₂Cu₃O_7-δ and Nd_{1+ x} Ba_{2- x} Cu₃O_7-δ for several doping levels. We isolated short-range dynamical charge density fluctuations in addition to the previously known quasi-critical charge density waves. They persist up to well above the pseudogap temperature T*, are characterized by energies of a few milli-electron volts, and pervade a large area of the phase diagram.

Spatially inhomogeneous competition between superconductivity and the charge density wave in YBa2Cu3O6.67

The charge density wave in the high-temperature superconductor YBa₂Cu₃O_7-x (YBCO) has two different ordering tendencies differentiated by their c-axis correlations. These correspond to ferro- (F-CDW) and antiferro- (AF-CDW) couplings between CDWs in neighbouring CuO₂ bilayers. This discovery has prompted several fundamental questions: how does superconductivity adjust to two competing orders and are either of these orders responsible for the electronic reconstruction? Here we use x-ray diffraction to study YBa₂Cu₃O_6.67 as a function of magnetic field and temperature. We show that regions with F-CDW correlations suppress superconductivity more strongly than those with AF-CDW correlations. This implies that an inhomogeneous superconducting state exists, in which some regions show a fragile form of superconductivity. By comparison of F-CDW and AF-CDW correlation lengths, it is concluded that F-CDW ordering is sufficiently long-range to modify the electronic structure. Our study thus suggests that F-CDW correlations impact both the superconducting and normal state properties of YBCO.

Distinction between pristine and disorder-perturbed charge density waves in ZrTe3

Charge density waves (CDWs) in the cuprate high-temperature superconductors have evoked much interest, yet their typical short-range nature has raised questions regarding the role of disorder. Here we report a resonant X-ray diffraction study of ZrTe[Formula: see text], a model CDW system, with focus on the influence of disorder. Near the CDW transition temperature, we observe two independent signals that arise concomitantly, only to become clearly separated in momentum while developing very different correlation lengths in the well-ordered state that is reached at a distinctly lower temperature. Anomalously slow dynamics of mesoscopic charge domains are further found near the transition temperature, in spite of the expected strong thermal fluctuations. Our observations signify the presence of distinct experimental fingerprints of pristine and disorder-perturbed CDWs. We discuss the latter also in the context of Friedel oscillations, which we argue might promote CDW formation via a self-amplifying process.

Charge ordering in high-temperature superconductors visualized by scanning tunneling microscopy

Since the discovery of stripe order in La_1.6-x Nd_0.4Sr _x CuO₄ superconductors in 1995, charge ordering in cuprate superconductors has been intensively studied by various experimental techniques. Among these studies, scanning tunneling microscope (STM) plays an irreplaceable role in determining the real space structures of charge ordering. STM imaging of different families of cuprates over a wide range of doping levels reveal similar checkerboard-like patterns, indicating that such a charge ordered state is likely a ubiquitous and intrinsic characteristic of cuprate superconductors, which may shed light on understanding the mechanism of high-temperature superconductivity. In another class of high-temperature superconductors, iron-based superconductors, STM studies reveal several charge ordered states as well, but their real-space patterns and the interplay with superconductivity are markedly different among different materials. In this paper, we present a brief review on STM studies of charge ordering in these two classes of high-temperature superconductors. Possible origins of charge ordering and its interplay with superconductivity will be discussed.

High-temperature superconductivity in monolayer Bi2Sr2CaCu2O8+δ

Although copper oxide high-temperature superconductors constitute a complex and diverse material family, they all share a layered lattice structure. This curious fact prompts the question of whether high-temperature superconductivity can exist in an isolated monolayer of copper oxide, and if so, whether the two-dimensional superconductivity and various related phenomena differ from those of their three-dimensional counterparts. The answers may provide insights into the role of dimensionality in high-temperature superconductivity. Here we develop a fabrication process that obtains intrinsic monolayer crystals of the high-temperature superconductor Bi₂Sr₂CaCu₂O_8+δ (Bi-2212; here, a monolayer refers to a half unit cell that contains two CuO₂ planes). The highest superconducting transition temperature of the monolayer is as high as that of optimally doped bulk. The lack of dimensionality effect on the transition temperature defies expectations from the Mermin-Wagner theorem, in contrast to the much-reduced transition temperature in conventional two-dimensional superconductors such as NbSe₂. The properties of monolayer Bi-2212 become extremely tunable; our survey of superconductivity, the pseudogap, charge order and the Mott state at various doping concentrations reveals that the phases are indistinguishable from those in the bulk. Monolayer Bi-2212 therefore displays all the fundamental physics of high-temperature superconductivity. Our results establish monolayer copper oxides as a platform for studying high-temperature superconductivity and other strongly correlated phenomena in two dimensions.