A four unit cell periodic pattern of quasi-particle states surrounding vortex cores in Bi2Sr2CaCu2O8+delta

Unidirectional Charge Orders Induced by Oxygen Vacancies on SrTiO3(001)

The discovery of high-mobility two-dimensional electron gas and low carrier density superconductivity in multiple SrTiO3-based heterostructures has stimulated intense interest in the surface properties of SrTiO3. The recent discovery of high-Tc superconductivity in the monolayer FeSe/SrTiO3 led to the upsurge and underscored the atomic precision probe of the surface structure. By performing atomically resolved cryogenic scanning tunneling microscopy/spectroscopy characterization on dual-TiO2-δ-terminated SrTiO3(001) surfaces with (√13 × √13), c(4 × 2), mixed (2 × 1), and (2 × 2) reconstructions, we disclosed universally broken rotational symmetry and contrasting bias- and temperature-dependent electronic states for apical and equatorial oxygen sites. With the sequentially evolved surface reconstructions and simultaneously increasing equatorial oxygen vacancies, the surface anisotropy reduces and the work function lowers. Intriguingly, unidirectional stripe orders appear on the c(4 × 2) surface, whereas local (4 × 4) order emerges and eventually forms long-range unidirectional c(4 × 4) charge order on the (2 × 2) surface. This work reveals robust unidirectional charge orders induced by oxygen vacancies due to strong and delicate electronic-lattice interaction under broken rotational symmetry, providing insights into understanding the complex behaviors in perovskite oxide-based heterostructures.

Variational Monte Carlo study of stripes as a function of doping in thet-t'Hubbard model

We perform variational Monte Carlo simulations of the single-band Hubbard model on the square lattice with both nearest (t) and next-nearest (t') neighbor hoppings. Our work investigates the consequences of increasing hole doping on the instauration of stripes and the behavior of the superconducting order parameter, with a discussion on how the two phenomena affect each other. We consider two different values of the next-nearest neighbor hopping parameter, that are appropriate for describing cuprate superconductors. We observe that stripes are the optimal state in a wide doping range; the stripe wavelength reduces at increasing doping, until stripes melt into a uniform state for large values of doping. Superconducting pair-pair correlations, indicating the presence of superconductivity, are always suppressed in the presence of stripes. Our results suggest that the phase diagram for the single-band Hubbard model is dominated by stripes, with superconductivity being possible only in a narrow doping range between striped states and a nonsuperconducting metal.

Tunable Non-Fermi Liquid Phase from Coupling to Two-Level Systems

We study a controlled large-N theory of electrons coupled to dynamical two-level systems (TLSs) via spatially random interactions. Such a physical situation arises when electrons scatter off low-energy excitations in a metallic glass, such as a charge or stripe glass. Our theory is governed by a non-Gaussian saddle point, which maps to the celebrated spin-boson model. By tuning the coupling strength we find that the model crosses over from a Fermi liquid at weak coupling to an extended region of non-Fermi liquid behavior at strong coupling, and realizes a marginal Fermi liquid at the crossover. Our results are valid for generic space dimensions d>1.

Vortex entropy and superconducting fluctuations in ultrathin underdoped Bi2Sr2CaCu2O8+x superconductor

Vortices in superconductors can help identify emergent phenomena but certain fundamental aspects of vortices, such as their entropy, remain poorly understood. Here, we study the vortex entropy in underdoped Bi2Sr2CaCu2O8+x by measuring both magneto-resistivity and Nernst effect on ultrathin flakes (≤2 unit-cell). We extract the London penetration depth from the magneto-transport measurements on samples with different doping levels. It reveals that the superfluid phase stiffness ρs scales linearly with the superconducting transition temperature Tc, down to the extremely underdoped case. On the same batch of ultrathin flakes, we measure the Nernst effect via on-chip thermometry. Together, we obtain the vortex entropy and find that it decays exponentially with Tc or ρs. We further analyze the Nernst signal above Tc in the framework of Gaussian superconducting fluctuations. The combination of electrical and thermoelectric measurements in the two-dimensional limit provides fresh insight into high temperature superconductivity.

Using strain to uncover the interplay between two- and three-dimensional charge density waves in high-temperature superconducting YBa2Cu3Oy

Uniaxial pressure provides an efficient approach to control charge density waves in YBa2Cu3Oy. It can enhance the correlation volume of ubiquitous short-range two-dimensional charge-density-wave correlations, and induces a long-range three-dimensional charge density wave, otherwise only accessible at large magnetic fields. Here, we use x-ray diffraction to study the strain dependence of these charge density waves and uncover direct evidence for a form of competition between them. We show that this interplay is qualitatively described by including strain effects in a nonlinear sigma model of competing superconducting and charge-density-wave orders. Our analysis suggests that strain stabilizes the 3D charge density wave in the regions between disorder-pinned domains of 2D charge density waves, and that the two orders compete at the boundaries of these domains. No signatures of discommensurations nor of pair density waves are observed. From a broader perspective, our results underscore the potential of strain tuning as a powerful tool for probing competing orders in quantum materials.

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.

Cryogenic spectroscopic imaging scanning tunnelling microscope in a water-cooled magnet down to 1.7 K

Spectroscopic-imaging scanning tunnelling microscope (SI-STM) in a water-cooled magnet (WM) at low temperature has long been desirable in the condensed matter physics area since it is crucial for addressing various scientific problems, such as the behaviour of Cooper electrons crossing Hc2 in a high-temperature superconductor. Here we report on the construction and performance of the first atomically resolved cryogenic SI-STM in a WM. It operates at low temperatures of down to 1.7 K and in magnetic fields of up to 22 T (the WM's upper safety limit). The WM-SI-STM unit features a high-stiffness sapphire-based frame with the lowest eigenfrequency being 16 kHz. A slender piezoelectric scan tube (PST) is coaxially embedded in and glued to the frame. A well-polished zirconia shaft is spring-clamped onto the gold-coated inner wall of the PST to serve both the stepper and the scanner. The microscope unit as a whole is elastically suspended in a tubular sample space inside a 1K-cryostat by a two-stage internal passive vibrational reduction system, achieving a base temperature below 2 K in a static exchange gas. We demonstrate the SI-STM by imaging TaS2 at 50 K and FeSe at 1.7 K. Detecting the well-defined superconducting gap of FeSe, an iron-based superconductor, at variable magnetic fields demonstrates the device's spectroscopic imaging capability. The maximum noise intensity at the typical frequency is 3 pA per square root Hz at 22 T, which is only slightly worse than at 0 T, indicating the insensitivity of the STM to harsh conditions. In addition, our work shows the potential of SI-STMs for use in a WM and hybrid magnet with a 50 mm-bore size where high fields can be generated.

Hyperbolic Fringe Signal for Twin Impurity Quasiparticle Interference

We study the quasiparticle interference (QPI) pattern emanating from a pair of adjacent impurities on the surface of a gapped superconductor (SC). We find that hyperbolic fringes (HFs) in the QPI signal can appear due to the loop contribution of the two-impurity scattering, where the locations of the two impurities are the hyperbolic focus points. For a single pocket Fermiology, a HF pattern signals chiral SC order for nonmagnetic impurities and requires magnetic impurities for a nonchiral SC. For a multipocket scenario, a sign-changing order parameter such as an s_{±} wave likewise yields a HF signature. We discuss twin impurity QPI as a new tool to complement the analysis of superconducting order from local spectroscopy.

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.

Charge order driven by multiple-Q spin fluctuations in heavily electron-doped iron selenide superconductors

Intertwined spin and charge orders have been widely studied in high-temperature superconductors, since their fluctuations may facilitate electron pairing; however, they are rarely identified in heavily electron-doped iron selenides. Here, using scanning tunneling microscopy, we show that when the superconductivity of (Li_0.84Fe_0.16OH)Fe_1-xSe is suppressed by introducing Fe-site defects, a short-ranged checkerboard charge order emerges, propagating along the Fe-Fe directions with an approximately 2a_Fe period. It persists throughout the whole phase space tuned by Fe-site defect density, from a defect-pinned local pattern in optimally doped samples to an extended order in samples with lower T_c or non-superconducting. Intriguingly, our simulations indicate that the charge order is likely driven by multiple-Q spin density waves originating from the spin fluctuations observed by inelastic neutron scattering. Our study proves the presence of a competing order in heavily electron-doped iron selenides, and demonstrates the potential of charge order as a tool to detect spin fluctuations.

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.

A spectroscopic-imaging scanning tunneling microscope in vector magnetic field

Cryogenic scanning tunneling microscopy and spectroscopy (STM/STS) performed in a high vector magnetic field provide unique possibilities for imaging surface magnetic structures and anisotropic superconductivity and exploring spin physics in quantum materials with atomic precision. Here, we describe the design, construction, and performance of a low-temperature, ultra-high-vacuum (UHV) spectroscopic-imaging STM equipped with a vector magnet capable of applying a field of up to 3 T in any direction with respect to the sample surface. The STM head is housed in a fully bakeable UHV compatible cryogenic insert and is operational over variable temperatures ranging from ∼300 down to 1.5 K. The insert can be easily upgraded using our home-designed ³He refrigerator. In addition to layered compounds, which can be cleaved at a temperature of either ∼300, ∼77, or ∼4.2 K to expose an atomically flat surface, thin films can also be studied by directly transferring using a UHV suitcase from our oxide thin-film laboratory. Samples can be treated further with a heater and a liquid helium/nitrogen cooling stage on a three-axis manipulator. The STM tips can be treated in vacuo by e-beam bombardment and ion sputtering. We demonstrate the successful operation of the STM with varying the magnetic field direction. Our facility provides a way to study materials in which magnetic anisotropy is a key factor in determining the electronic properties such as in topological semimetals and superconductors.

The magnetic field driven superconductor-metal transition in disordered hole-overdoped cuprates

By solving the Bogoliubov-de Gennes equations for ad-wave superconductor, we explore how the interplay between disorder and the orbital depairing of an external magnetic field influences the superconductor-metal transition of the hole-overdoped cuprates. For highly disordered systems, we find granular Cooper paring to persist above the critical field where the superfluid stiffness goes to zero. We also show that because the vortices are attracted to regions where the superconducting pairing is already weak, the Caroli-de Gennes-Matricon zero-bias peak in the local density of states at the vortex cores disappears already at moderate disorder.

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.

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.

Enhanced charge density wave coherence in a light-quenched, high-temperature superconductor

Superconductivity and charge density waves (CDWs) are competitive, yet coexisting, orders in cuprate superconductors. To understand their microscopic interdependence, a probe capable of discerning their interaction on its natural length and time scale is necessary. We use ultrafast resonant soft x-ray scattering to track the transient evolution of CDW correlations in YBa₂Cu₃O_6+x after the quench of superconductivity by an infrared laser pulse. We observe a nonthermal response of the CDW order characterized by a near doubling of the correlation length within ≈1 picosecond of the superconducting quench. Our results are consistent with a model in which the interaction between superconductivity and CDWs manifests inhomogeneously through disruption of spatial coherence, with superconductivity playing the dominant role in stabilizing CDW topological defects, such as discommensurations.

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.

Direct visualization of a static incommensurate antiferromagnetic order in Fe-doped Bi2Sr2CaCu2O8+δ

In cuprate superconductors, due to strong electronic correlations, there are multiple intertwined orders which either coexist or compete with superconductivity. Among them, the antiferromagnetic (AF) order is the most prominent one. In the region where superconductivity sets in, the long-range AF order is destroyed. Yet the residual short-range AF spin fluctuations are present up to a much higher doping, and their role in the emergence of the superconducting phase is still highly debated. Here, by using a spin-polarized scanning tunneling microscope, we directly visualize an emergent incommensurate AF order in the nearby region of Fe impurities embedded in the optimally doped Bi₂Sr₂CaCu₂O_8+δ (Bi2212). Remarkably, the Fe impurities suppress the superconducting coherence peaks with the gapped feature intact, but pin down the ubiquitous short-range incommensurate AF order. Our work shows an intimate relation between antiferromagnetism and superconductivity.

Topological Doping and Superconductivity in Cuprates: An Experimental Perspective

Hole doping into a correlated antiferromagnet leads to topological stripe correlations, involving charge stripes that separate antiferromagnetic spin stripes of opposite phases. The topological spin stripe order causes the spin degrees of freedom within the charge stripes to feel a geometric frustration with their environment. In the case of cuprates, where the charge stripes have the character of a hole-doped two-leg spin ladder, with corresponding pairing correlations, anti-phase Josephson coupling across the spin stripes can lead to a pair-density-wave order in which the broken translation symmetry of the superconducting wave function is accommodated by pairs with finite momentum. This scenario is now experimentally verified by recently reported measurements on La2−xBaxCuO4 with x=1/8. While pair-density-wave order is not common as a cuprate ground state, it provides a basis for understanding the uniform d-wave order that is more typical in superconducting cuprates.

Imaging the atomic-scale electronic states induced by a pair of hole dopants in Ca2CuO2Cl2 Mott insulator

We use scanning tunneling microscopy to visualize the atomic-scale electronic states induced by a pair of hole dopants in Ca₂CuO₂Cl₂ parent Mott insulator of cuprates. We find that when the two dopants approach each other, the transfer of spectral weight from high energy Hubbard band to low energy in-gap state creates a broad peak and nearly V-shaped gap around the Fermi level. The peak position shows a sudden drop at distance around 4 a₀ and then remains almost constant. The in-gap states exhibit peculiar spatial distributions depending on the configuration of the two dopants relative to the underlying Cu lattice. These results shed important new lights on the evolution of low energy electronic states when a few holes are doped into parent cuprates.

Multiple Charge Density Waves and Superconductivity Nucleation at Antiphase Domain Walls in the Nematic Pnictide Ba_{1-x}Sr_{x}Ni_{2}As_{2}

How superconductivity interacts with charge or nematic order is one of the great unresolved issues at the center of research in quantum materials. Ba_{1-x}Sr_{x}Ni_{2}As_{2} (BSNA) is a charge ordered pnictide superconductor recently shown to exhibit a sixfold enhancement of superconductivity due to nematic fluctuations near a quantum phase transition (at x_{c}=0.7) [1]. The superconductivity is, however, anomalous, with the resistive transition for 0.4<x<x_{c} occurring at a higher temperature than the specific heat anomaly. Using x-ray scattering, we discovered a new charge density wave (CDW) in BSNA in this composition range. The CDW is commensurate with a period of two lattice parameters, and is distinct from the two CDWs previously reported in this material [1,2]. We argue that the anomalous transport behavior arises from heterogeneous superconductivity nucleating at antiphase domain walls in this CDW. We also present new data on the incommensurate CDW, previously identified as being unidirectional [2], showing that it is a rotationally symmetric "4Q" state with C_{4} symmetry. Our study establishes BSNA as a rare material containing three distinct CDWs, and an exciting test bed for studying coupling between CDW, nematic, and SC orders.

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.

Charge Condensation and Lattice Coupling Drives Stripe Formation in Nickelates

Revealing the predominant driving force behind symmetry breaking in correlated materials is sometimes a formidable task due to the intertwined nature of different degrees of freedom. This is the case for La_{2-x}Sr_{x}NiO_{4+δ}, in which coupled incommensurate charge and spin stripes form at low temperatures. Here, we use resonant x-ray photon correlation spectroscopy to study the temporal stability and domain memory of the charge and spin stripes in La_{2-x}Sr_{x}NiO_{4+δ}. Although spin stripes are more spatially correlated, charge stripes maintain a better temporal stability against temperature change. More intriguingly, charge order shows robust domain memory with thermal cycling up to 250 K, far above the ordering temperature. These results demonstrate the pinning of charge stripes to the lattice and that charge condensation is the predominant factor in the formation of stripe orders in nickelates.

Incommensurate smectic phase in close proximity to the high-Tc superconductor FeSe/SrTiO3

Superconductivity is significantly enhanced in monolayer FeSe grown on SrTiO₃, but not for multilayer films, in which large strength of nematicity develops. However, the link between the high-transition temperature superconductivity in monolayer and the correlation related nematicity in multilayer FeSe films is not well understood. Here, we use low-temperature scanning tunneling microscopy to study few-layer FeSe thin films grown by molecular beam epitaxy. We observe an incommensurate long-range smectic phase, which solely appears in bilayer FeSe films. The smectic order still locally exists and gradually fades away with increasing film thickness, while it suddenly vanishes in monolayer FeSe, indicative of an abrupt smectic phase transition. Surface alkali-metal doping can suppress the smectic phase and induce high-T_c superconductivity in bilayer FeSe. Our observations provide evidence that the monolayer FeSe is in close proximity to the smectic phase, and its superconductivity is likely enhanced by this electronic instability as well.

The relation between enhanced superconductivity in monolayer FeSe grown on SrTiO₃ and the large nematicity in multilayer FeSe on SrTiO₃ remains not well understood. Here, the authors observe a long-range smectic phase in bilayer FeSe films but vanishes in monolayer FeSe, providing a new instability to help enhance the superconductivity.

Observation of Distinct Spatial Distributions of the Zero and Nonzero Energy Vortex Modes in (Li_{0.84}Fe_{0.16})OHFeSe

The energy and spatial distributions of vortex bound state in superconductors carry important information about superconducting pairing and the electronic structure. Although discrete vortex states, and sometimes a zero energy mode, had been observed in several iron-based superconductors, their spatial properties are rarely explored. In this study, we used low-temperature scanning tunneling microscopy to measure the vortex state of (Li,Fe)OHFeSe with high spatial resolution. We found that the nonzero energy states display clear spatial oscillations with a period corresponding to bulk Fermi wavelength; while in contrast, the zero energy mode does not show such oscillation, which suggests its distinct electronic origin. Furthermore, the oscillations of positive and negative energy states near E_{F} are found to be clearly out of phase. Based on a two-band model calculation, we show that our observation is more consistent with an s_{++} wave pairing in the bulk of (Li, Fe)OHFeSe, and superconducting topological states on the surface.

Observation of an electronic order along [110] direction in FeSe

Multiple ordered states have been observed in unconventional superconductors. Here, we apply scanning tunneling microscopy to probe the intrinsic ordered states in FeSe, the structurally simplest iron-based superconductor. Besides the well-known nematic order along [100] direction, we observe a checkerboard charge order in the iron lattice, which we name a [110] electronic order in FeSe. The [110] electronic order is robust at 77 K, accompanied with the rather weak [100] nematic order. At 4.5 K, The [100] nematic order is enhanced, while the [110] electronic order forms domains with reduced correlation length. In addition, the collective [110] order is gaped around [−40, 40] meV at 4.5 K. The observation of this exotic electronic order may shed new light on the origin of the ordered states in FeSe.

Understanding the relation of different electronic orders in high temperature superconductors is of fundamental interest. Here, the authors observe a checkerboard charge order along [110] direction of FeSe.

On the Kinetic Energy Driven Superconductivity in the Two-Dimensional Hubbard Model

We investigate the role of kinetic energy for the stability of superconducting state in the two-dimensional Hubbard model on the basis of an optimization variational Monte Carlo method. The wave function is optimized by multiplying by correlation operators of site off-diagonal type. This wave function is written in an exponential-type form given as ψλ=exp(−λK)ψG for the Gutzwiller wave function ψG and a kinetic operator K. The kinetic correlation operator exp(−λK) plays an important role in the emergence of superconductivity in large-U region of the two-dimensional Hubbard model, where U is the on-site Coulomb repulsive interaction. We show that the superconducting condensation energy mainly originates from the kinetic energy in the strongly correlated region. This may indicate a possibility of high-temperature superconductivity due to the kinetic energy effect in correlated electron systems.

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.

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.

Competing stripe and magnetic phases in the cuprates from first principles

Realistic description of competing phases in complex quantum materials has proven extremely challenging. For example, much of the existing density-functional-theory-based first-principles framework fails in the cuprate superconductors. Various many-body approaches involve generic model Hamiltonians and do not account for the interplay between the spin, charge, and lattice degrees of freedom. Here, by deploying the recently constructed strongly constrained and appropriately normed (SCAN) density functional, we show how the landscape of competing stripe and magnetic phases can be addressed on a first-principles basis both in the parent insulator YBa2Cu3O6 and the near-optimally doped YBa2Cu3O7 as archetype cuprate compounds. In YBa2Cu3O7, we find many stripe phases that are nearly degenerate with the ground state and may give rise to the pseudogap state from which the high-temperature superconducting state emerges. We invoke no free parameters such as the Hubbard U, which has been the basis of much of the existing cuprate literature. Lattice degrees of freedom are found to be crucially important in stabilizing the various phases.

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.

Anomalies in the pseudogap phase of the cuprates: competing ground states and the role of umklapp scattering

Over the past two decades, advances in computational algorithms have revealed a curious property of the two-dimensional Hubbard model (and related theories) with hole doping: the presence of close-in-energy competing ground states that display very different physical properties. On the one hand, there is a complicated state exhibiting intertwined spin, charge, and pair density wave orders. We call this 'type A'. On the other hand, there is a uniform d-wave superconducting state that we denote as 'type B'. We advocate, with the support of both microscopic theoretical calculations and experimental data, dividing the high-temperature cuprate superconductors into two corresponding families, whose properties reflect either the type A or type B ground states at low temperatures. We review the anomalous properties of the pseudogap phase that led us to this picture, and present a modern perspective on the role that umklapp scattering plays in these phenomena in the type B materials. This reflects a consistent framework that has emerged over the last decade, in which Mott correlations at weak coupling drive the formation of the pseudogap. We discuss this development, recent theory and experiments, and open issues.

Magnetic field-induced pair density wave state in the cuprate vortex halo

High magnetic fields suppress cuprate superconductivity to reveal an unusual density wave (DW) state coexisting with unexplained quantum oscillations. Although routinely labeled a charge density wave (CDW), this DW state could actually be an electron-pair density wave (PDW). To search for evidence of a field-induced PDW, we visualized modulations in the density of electronic states N(r) within the halo surrounding Bi₂Sr₂CaCu₂O₈ vortex cores. We detected numerous phenomena predicted for a field-induced PDW, including two sets of particle-hole symmetric N(r) modulations with wave vectors Q_P and 2Q _P , with the latter decaying twice as rapidly from the core as the former. These data imply that the primary field-induced state in underdoped superconducting cuprates is a PDW, with approximately eight CuO₂ unit-cell periodicity and coexisting with its secondary CDWs.

Observation of two types of charge-density-wave orders in superconducting La2-xSrxCuO4

The discovery of charge- and spin-density-wave (CDW/SDW) orders in superconducting cuprates has altered our perspective on the nature of high-temperature superconductivity (SC). However, it has proven difficult to fully elucidate the relationship between the density wave orders and SC. Here, using resonant soft X-ray scattering, we study the archetypal cuprate La_2-xSr_xCuO₄ (LSCO) over a broad doping range. We reveal the existence of two types of CDW orders in LSCO, namely CDW stripe order and CDW short-range order (SRO). While the CDW-SRO is suppressed by SC, it is partially transformed into the CDW stripe order with developing SDW stripe order near the superconducting T_c. These findings indicate that the stripe orders and SC are inhomogeneously distributed in the superconducting CuO₂ planes of LSCO. This further suggests a new perspective on the putative pair-density-wave order that coexists with SC, SDW, and CDW orders.

To fully elucidate the relationship between density wave orders and superconductivity in high-T_c cuprates remains difficult. Here, the authors reveal two types of charge-density-wave orders and their intertwined relationship with spin-density-wave order and superconductivity in La_2-xSr_xCuO_4.

Mechanism of High-Temperature Superconductivity in Correlated-Electron Systems

It is very important to elucidate the mechanism of superconductivity for achieving room temperature superconductivity. In the first half of this paper, we give a brief review on mechanisms of superconductivity in many-electron systems. We believe that high-temperature superconductivity may occur in a system with interaction of large-energy scale. Empirically, this is true for superconductors that have been found so far. In the second half of this paper, we discuss cuprate high-temperature superconductors. We argue that superconductivity of high temperature cuprates is induced by the strong on-site Coulomb interaction, that is, the origin of high-temperature superconductivity is the strong electron correlation. We show the results on the ground state of electronic models for high temperature cuprates on the basis of the optimization variational Monte Carlo method. A high-temperature superconducting phase will exist in the strongly correlated region.

Uniaxial pressure control of competing orders in a high-temperature superconductor

Cuprates exhibit antiferromagnetic, charge density wave (CDW), and high-temperature superconducting ground states that can be tuned by means of doping and external magnetic fields. However, disorder generated by these tuning methods complicates the interpretation of such experiments. Here, we report a high-resolution inelastic x-ray scattering study of the high-temperature superconductor YBa₂Cu₃O_6.67 under uniaxial stress, and we show that a three-dimensional long-range-ordered CDW state can be induced through pressure along the a axis, in the absence of magnetic fields. A pronounced softening of an optical phonon mode is associated with the CDW transition. The amplitude of the CDW is suppressed below the superconducting transition temperature, indicating competition with superconductivity. The results provide insights into the normal-state properties of cuprates and illustrate the potential of uniaxial-pressure control of competing orders in quantum materials.

Selection Rules for Quasiparticle Interference with Internal Nonsymmorphic Symmetries

We study how nonsymmorphic symmetries that commute with lattice translations are reflected in the quasiparticle interference (QPI) maps measured by scanning tunneling microscopy (STM). QPI maps, which result from scattering of Bloch states off impurities, record the interference of incoming and scattered waves as a function of energy and tip's position. Although both the impurity and the tip generically break spatial symmetries, we find that the QPI maps provide universal information on these symmetries. The symmetries impose constraints on the relation between various momentum components of the Bloch functions. These relations result in selection rules on certain momentum transfers in QPI maps. We find that universal information is encoded in the absence of QPI signal, or in the relative intensity of its replications. We show examples for one-dimensional chains and an effective model of the layered compound ZrSiS. We discuss the implications of our theory in the analysis of observed QPI of the Weyl semimetal TaAs. Our theory is particularly relevant for materials in rod and layer space groups, or when a correlated order parameter, such as antiferromagnetism, enlarges the unit cell.

Two-Dimensional Band Structure in Honeycomb Metal-Organic Frameworks

Two-dimensional (2D) metal-organic frameworks (MOFs) have been recently proposed as a flexible material platform for realizing exotic quantum phases including topological and anomalous quantum Hall insulators. Experimentally, direct synthesis of 2D MOFs has been essentially confined to metal substrates, where the strong interaction with the substrate masks the intrinsic electronic properties of the MOF. In addition to electronic decoupling from the underlying metal support, synthesis on weakly interacting substrates (e.g., graphene) would enable direct realization of heterostructures of 2D MOFs with inorganic 2D materials. Here, we demonstrate synthesis of 2D honeycomb MOFs on epitaxial graphene substrate. Using low-temperature scanning tunneling microscopy (STM) and atomic force microscopy (AFM) complemented by density-functional theory (DFT) calculations, we show the formation of a 2D band structure in the MOF decoupled from the substrate. These results open the experimental path toward MOF-based designer electronic materials with complex, engineered electronic structures.

A scanning tunneling microscope for spectroscopic imaging below 90 mK in magnetic fields up to 17.5 T

We describe the development and performance of an ultra-high vacuum scanning tunneling microscope working under combined extreme conditions of ultra-low temperatures and high magnetic fields. We combined a top-loading dilution refrigerator and a standard bucket dewar with a bottom-loading superconducting magnet to achieve 4.5 days operating time, which is long enough to perform various spectroscopic-imaging measurements. To bring the effective electron temperature closer to the mixing-chamber temperature, we paid particular attention to filtering out radio-frequency noise, as well as enhancing the thermal link between the microscope unit and the mixing chamber. We estimated the lowest effective electron temperature to be below 90 mK by measuring the superconducting-gap spectrum of aluminum. We confirmed the long-term stability of the spectroscopic-imaging measurement by visualizing superconducting vortices in the cuprate superconductor Bi₂Sr₂CaCu₂O_8+δ .

Pseudo-spin skyrmions in the phase diagram of cuprate superconductors

Topological states of matter are at the root of some of the most fascinating phenomena in condensed matter physics. Here we argue that skyrmions in the pseudo-spin space related to an emerging SU(2) symmetry enlighten many mysterious properties of the pseudogap phase in under-doped cuprates. We detail the role of the SU(2) symmetry in controlling the phase diagram of the cuprates, in particular how a cascade of phase transitions explains the arising of the pseudogap, superconducting and charge modulation phases seen at low temperature. We specify the structure of the charge modulations inside the vortex core below T _c, as well as in a wide temperature region above T _c, which is a signature of the skyrmion topological structure. We argue that the underlying SU(2) symmetry is the main structure controlling the emergent complexity of excitations at the pseudogap scale T ^*. The theory yields a gapping of a large part of the anti-nodal region of the Brillouin zone, along with q  =  0 phase transitions, of both nematic and loop currents characters.

Discrete energy levels of Caroli-de Gennes-Matricon states in quantum limit in FeTe0.55Se0.45

Caroli-de Gennes-Matricon (CdGM) states were predicted in 1964 as low-energy excitations within vortex cores of type-II superconductors. In the quantum limit, the energy levels of these states were predicted to be discrete with the basic levels at ±μΔ2/EF (μ = 1/2, 3/2, 5/2, …) with Δ the superconducting energy gap and EF the Fermi energy. However, due to the small ratio of Δ/EF in most type-II superconductors, it is very difficult to observe the discrete CdGM states, but rather a symmetric peak which appears at zero bias at the vortex center. Here we report the clear observation of these discrete energy levels of CdGM states in FeTe0.55Se0.45. The rather stable energies of these bound state peaks vs. space clearly validate our conclusion. Analysis based on the energies of these CdGM states indicates that the Fermi energy in the present system is very small.

Visualizing heavy fermion confinement and Pauli-limited superconductivity in layered CeCoIn5

Layered material structures play a key role in enhancing electron–electron interactions to create correlated metallic phases that can transform into unconventional superconducting states. The quasi-two-dimensional electronic properties of such compounds are often inferred indirectly through examination of bulk properties. Here we use scanning tunneling microscopy to directly probe in cross-section the quasi-two-dimensional electronic states of the heavy fermion superconductor CeCoIn₅. Our measurements reveal the strong confined nature of quasiparticles, anisotropy of tunneling characteristics, and layer-by-layer modulated behavior of the precursor pseudogap gap phase. In the interlayer coupled superconducting state, the orientation of line defects relative to the d-wave order parameter determines whether in-gap states form due to scattering. Spectroscopic imaging of the anisotropic magnetic vortex cores directly characterizes the short interlayer superconducting coherence length and shows an electronic phase separation near the upper critical in-plane magnetic field, consistent with a Pauli-limited first-order phase transition into a pseudogap phase.

The electronic properties along the out-of-plane direction of layered materials are often inferred indirectly. Here, Gyenis et al. directly probe in cross-section the quasi-two-dimensional correlated electronic states of the heavy fermion superconductor CeCoIn₅.

Ultrafast current imaging by Bayesian inversion

Spectroscopic measurements of current–voltage curves in scanning probe microscopy is the earliest and one of the most common methods for characterizing local energy-dependent electronic properties, providing insight into superconductive, semiconductor, and memristive behaviors. However, the quasistatic nature of these measurements renders them extremely slow. Here, we demonstrate a fundamentally new approach for dynamic spectroscopic current imaging via full information capture and Bayesian inference. This general-mode I–V method allows three orders of magnitude faster measurement rates than presently possible. The technique is demonstrated by acquiring I–V curves in ferroelectric nanocapacitors, yielding >100,000 I–V curves in <20 min. This allows detection of switching currents in the nanoscale capacitors, as well as determination of the dielectric constant. These experiments show the potential for the use of full information capture and Bayesian inference toward extracting physics from rapid I–V measurements, and can be used for transport measurements in both atomic force and scanning tunneling microscopy.

Scanning probe microscopy is widely used to characterize material properties with atomic resolution, yet electronic property mapping is normally constrained by slow data acquisition. Somnath et al. show a current–voltage method, which enables fast electronic spectroscopy mapping over micrometer-sized areas.

Quasiparticle conductance-voltage characteristics for break junctions involving d-wave superconductors: charge-density-wave effects

Quasiparticle tunnel conductance-voltage characteristics (CVCs), [Formula: see text], were calculated for break junctions (BJs) made up of layered d-wave superconductors partially gapped by charge-density waves (CDWs). The current is assumed to flow in the ab-plane of electrodes. The influence of CDWs is analyzed by comparing the resulting CVCs with CVCs calculated for BJs made up of pure d-wave superconductors with relevant parameters. The main CDW-effects were found to be the appearance of new CVC peculiarities and the loss of CVC symmetry with respect to the V-sign. Tunnel directionality was shown to be one of the key factors in the formation of [Formula: see text] dependences. In particular, the orientation of electrodes with respect to the current channel becomes very important. As a result, [Formula: see text] can acquire a large variety of forms similar to those for tunnel junctions between superconductors with s-wave, d-wave, and mixed symmetry of their order parameters. The diversity of peculiarities is especially striking at finite temperatures. In the case of BJs made up of pure d-wave superconductors, the resulting CVC can include a two-peak gap-driven structure. The results were compared with the experimental BJ data for a number of high-T _c oxides. It was shown that the large variety of the observed current-voltage characteristics can be interpreted in the framework of our approach. Thus, quasiparticle tunnel currents in the ab-plane can be used as an additional mean to detect CDWs competing with superconductivity in cuprates or other layered superconductors.

Observation of Caroli-de Gennes-Matricon Vortex States in YBa_{2}Cu_{3}O_{7-δ}

The copper oxides present the highest superconducting temperature and properties at odds with other compounds, suggestive of a fundamentally different superconductivity. In particular, the Abrikosov vortices fail to exhibit localized states expected and observed in all clean superconductors. We have explored the possibility that the elusive vortex-core signatures are actually present but weak. Combining local tunneling measurements with large-scale theoretical modeling, we positively identify the vortex states in YBa_{2}Cu_{3}O_{7-δ}. We explain their spectrum and the observed variations thereof from one vortex to the next by considering the effects of nearby vortices and disorder in the vortex lattice. We argue that the superconductivity of copper oxides is conventional, but the spectroscopic signature does not look so because the superconducting carriers are a minority.

Bending and breaking of stripes in a charge ordered manganite

In charge-ordered phases, broken translational symmetry emerges from couplings between charge, spin, lattice, or orbital degrees of freedom, giving rise to remarkable phenomena such as colossal magnetoresistance and metal-insulator transitions. The role of the lattice in charge-ordered states remains particularly enigmatic, soliciting characterization of the microscopic lattice behavior. Here we directly map picometer scale periodic lattice displacements at individual atomic columns in the room temperature charge-ordered manganite Bi_0.35Sr_0.18Ca_0.47MnO₃ using aberration-corrected scanning transmission electron microscopy. We measure transverse, displacive lattice modulations of the cations, distinct from existing manganite charge-order models. We reveal locally unidirectional striped domains as small as ~5 nm, despite apparent bidirectionality over larger length scales. Further, we observe a direct link between disorder in one lattice modulation, in the form of dislocations and shear deformations, and nascent order in the perpendicular modulation. By examining the defects and symmetries of periodic lattice displacements near the charge ordering phase transition, we directly visualize the local competition underpinning spatial heterogeneity in a complex oxide.

Charge-density-wave order takes over antiferromagnetism in Bi2Sr2-x La x CuO6 superconductors

Superconductivity appears in the cuprates when a spin order is destroyed, while the role of charge is less known. Recently, charge density wave (CDW) was found below the superconducting dome in YBa₂Cu₃O_y when a high magnetic field is applied perpendicular to the CuO₂ plane, which was suggested to arise from incipient CDW in the vortex cores that becomes overlapped. Here by ⁶³Cu-nuclear magnetic resonance, we report the discovery of CDW induced by an in-plane field, setting in above the dome in single-layered Bi₂Sr_2−xLa_xCuO₆. The onset temperature T_CDW takes over the antiferromagnetic order temperature T_N beyond a critical doping level at which superconductivity starts to emerge, and scales with the pseudogap temperature T*. These results provide important insights into the relationship between spin order, CDW and the pseudogap, and their connections to high-temperature superconductivity.

Whilst superconductivity usually appears when magnetic order is suppressed, the role of charge is less known. Here, Kawasaki et al. report a charge density wave (CDW) above the superconducting transition induced by an in-plane magnetic field in Bi₂Sr_2-xLa_xCuO₆, with the CDW onset temperature scaling with the pseudogap temperature.