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At least 199 records · Page 11

Mechanical Behavior of Hybrids with Hyper-Connected Molecular Networks

Our research is focused on fundamental studies related to the molecular design, synthesis, characterization and modeling of molecular-reinforced hybrid glass films for superior mechanical and fracture properties. Molecular-reinforced hybrid glass films exhibit unique electro-optical properties while maintaining excellent thermal stability. They have important technological application for emerging nanoscience and energy technologies including anti-reflective and ultra-barrier layers in photovoltaics, display technologies, as well as quantum technologies. However, they are often inherently brittle in nature, do not adhere well to adjacent substrates, and exhibit poor mechanical properties. Furthermore, a robust understanding of the underlying mechanisms of deformation, fracture and fatigue has been lacking particularly given their mechanically fragile nature. The related lack of structure-property relationships between molecular structure, mechanical properties and processing represents a fundamental challenge to their integration into device technologies, compromises their thermomechanical reliability and has thwarted efforts to improve mechanical and, particularly, fracture properties.

36 MATERIALS SCIENCE↗

Ultrafast formation of a transient two-dimensional diamondlike structure in twisted bilayer graphene

Due to the absence of matching carbon atoms at honeycomb centers with carbon atoms in adjacent graphene sheets, theorists predicted that a sliding process is needed to form AA, AB', or ABC stacking when directly converting graphite into sp 3 bonded diamond. Here, using twisted bilayer graphene, which naturally provides AA and AB' stacking configurations, we report the ultrafast formation of a transient two-dimensional diamondlike structure (which is not observed in aligned graphene) under femtosecond laser irradiation. This photoinduced phase transition is evidenced by the appearance of bond lengths of 1.94 and 3.14 Å in the time-dependent differential pair distribution function using MeV ultrafast electron diffraction. Molecular dynamics and first-principles calculation indicate that sp 3 bonds nucleate at AA and AB' stacked areas in a moiré pattern. This work sheds light on the direct graphite-to-diamond transformation mechanism, which has not been fully understood for more than 60 years.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Parameter-free treatment of a layered correlated van der Waals magnet: CrPS 4

The electronic and magnetic structure of CrPS 4 , a two-dimensional (2D) magnetic semiconductor is examined by employing the SCAN meta-GGA density functional. We find the resulting magnetic moment and band gap are in excellent agreement with experiment. From the bulk magnetic configurations, we confirm the experimentally observed A-type antiferromagnetic (A-AFM) ordered ground state with a magnetic moment of 2.78 µ B per chromium atom and band gap of 1.34 eV. To gain insight into the evolution of the ground state with layers, the total energy of each magnetic configuration is calculated for a variety of thicknesses. Monolayer CrPS 4 is predicted to be a ferromagnetic insulator with a band gap of 1.37 eV, and A-AFM for bilayer and trilayer, with band gaps of 1.35 and 1.30 eV, respectively. The electronic structure is reported for the single-, two-, and three-layer and bulk CrPS 4 . Finally, we explore the optical properties of the 2D structure and report the dielectric tensor components and Kerr parameters for the monolayer.

2-dimensional systems↗

M PX 3 van der Waals magnets under pressure ( M = Mn, Ni, V, Fe, Co, Cd; X = S, Se)

van der Waals antiferromagnets with chemical formula MPX 3 (M = V, Mn, Fe, Co, Ni, Cd; X = S, Se) are superb platforms for exploring the fundamental properties of complex chalcogenides, revealing their structure-property relations and unraveling the physics of confinement. Pressure is extremely effective as an external stimulus, able to tune properties and drive new states of matter. In this review, we summarize experimental and theoretical progress to date with special emphasis on the structural, magnetic, and optical properties of the MPX 3 family of materials. Under compression, these compounds host inter-layer sliding and insulator-to-metal transitions accompanied by dramatic volume reduction and spin state collapse, piezochromism, possible polar metal and orbital Mott phases, as well as superconductivity. Some responses are already providing the basis for spintronic, magneto-optic, and thermoelectric devices. We propose that strain may drive similar functionality in these materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

The Li–F–H ternary system at high pressures

Evolutionary crystal structure prediction searches have been employed to explore the ternary Li-F-H system at 300 GPa. Metastable phases were uncovered within the static lattice approximation, with LiF 3 H 2 , LiF 2 H, Li 3 F 4 H, LiF 4 H 4 , Li 2 F 3 H and LiF 3 H lying within 50 meV/atom of the 0 K convex hull. All of these phases contain H n F¯ n+1 (n = 1; 2) anions, and Li + cations. Other structural motifs such as LiF slabs, $H$ $^{+}_{3}$ molecules and F δ- ions are present in some of the low enthalpy Li-F-H structures. The bonding within the H n F¯ n+1 molecules, which may be bent or linear, symmetric or asymmetric, is analyzed. The five phases closest to the hull are insulators, while LiF 3 H is metallic and predicted to have a vanishingly small superconducting critical temperature. Li 3 F 4 H is predicted to be stable at zero pressure. Furthermore, this study lays the foundation for future investigations of the role of temperature and anharmonicity on the stability and properties of compounds and alloys in the Li-F-H ternary system.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Suppression of antiferromagnetic order and strong ferromagnetic spin fluctuations in Ca(Co 1-x Ni x ) 2-y As 2 single crystals

CaCo 2–y As 2 is a unique itinerant system having strong magnetic frustration. Here, we report the effect of electron doping on the physical properties resulting from Ni substitutions for Co. The single crystals of Ca(Co 1–x Ni x ) 2–y As 2 were characterized by single-crystal x-ray diffraction, energy-dispersive x-ray spectroscopy, magnetization M versus temperature T, magnetic field H, time t, and heat capacity C p (H, T) measurements. The A-type antiferromagnetic (AFM) transition temperature T N = 52 K for x = 0 decreases to 22 K with only 3% Ni substitution and is completely suppressed for x > 0.16. For 0.11 ≤ x ≤ 0.52 strong ferromagnetic (FM) fluctuations develop as revealed by magnetic susceptibility χ(T) = M(T) / H measurements. For x = 0.11 and 0.16 competing AFM and FM interactions result in a reentrant spin-glass behavior below T N , as evidenced by the observations of thermomagnetic hysteresis and magnetic relaxation. Enhanced FM fluctuations are also found for the x = 0.21 and 0.31 crystals, where χ c increases significantly at low T. A large χ anisotropy in these compositions where χ c is up to a factor of two larger than χ ab suggests that the FM spin fluctuations are quasi-1D in nature. Weak ferromagnetic contributions to the magnetization are found at T = 2 K for x = 0.11 –0.31. Heat-capacity Cp(T) measurements reveal the presence of FM quantum spin fluctuations for 0.11 ≤ x ≤ 0.52, where a logarithmic T dependence of Cp(T) / T is observed at low T. Here, the suppression of AFM order by the development of strong FM fluctuations in Ca(Co 1–x Ni x ) 2–y As 2 crystals suggests the presence of a FM quantum-critical point at x ≈ 0.20. Our density-functional theory (DFT) calculations confirm that FM fluctuations are enhanced by Ni substitutions for Co in CaCo 2–y As 2 . The Sommerfeld electronic heat-capacity coefficient is enhanced for x = 0, 0.21, and 0.42 by about a factor of two compared to DFT calculations of the density of states (DOS) at the Fermi energy, suggesting an enhancement of the DOS from electron-phonon and/or electron-electron interactions. The crystals with x > 0.52 do not exhibit FM spin fluctuations or magnetic order at T ≥ 1.8 K, which was found from the DFT calculations to arise from a Stoner transition. Superconductivity is not observed above 1.8 K for any of the compositions. Neutron-diffraction studies of crystals with x = 0.11 and 0.16 in the crossover regime (0.1 ≲ x ≲ 0.2) show no evidence of A-type ordering as observed in the parent compound with x = 0. Furthermore, no other common magnetic structures, such as ferromagnetic (FM), helical stacking of in-plane FM layers, or in-plane AFM structure, are found with an ordered moment greater than the uncertainty of 0.05 μ B per transition-metal atom.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Structure-property correlations and scaling in the magnetic and magnetocaloric properties of GdCrO 3 particles

The structure, magnetic, and magnetocaloric (MC) properties of orthorhombic nanocrystalline GdCrO 3 with six particle sizes: < d > = 87, 103, 145, 224, 318, and 352 nm are reported. The particle size was tailored by annealing under different temperatures and estimated by scanning electron microscopy. With increase in < d >, Goldschmidt tolerance factor t, orthorhombic strain s, and out-of-plane Cr–O 1 –Cr bond angle first decrease, reaching minimum values for < d > = 224 nm, and then increase for sample with < d > = 318 nm and 352 nm, thus showing a V-shaped variation. Temperature dependence of the magnetization (M) reveals an antiferromagnetic transition at ${T}_{\text{N}}^{\text{Cr}}\sim 168$ K for < d > ≥ 224 nm and ${T}_{\text{N}}^{\text{Cr}}\sim 167$ K for < d > < 224 nm and an essentially d-independent spin-reorientation at T SR = 9 K. M measured at 5 K and 7 T first increases with increase in < d >, reaching maximum value for sample with < d > = 224 nm, and then decreases for samples with < d > = 318 nm and 352 nm, showing an inverted-V variation with < d >. Similar < d >-dependence is observed for the magnetic entropy change (MEC) and relative cooling power (RCP) showing a close relationship between the structural and magnetic properties of GdCrO 3 nanoparticles investigated here. The 224 nm sample with the minimum values of t, s, and Cr–O 1 –Cr bond angle exhibits the maximum value of MEC (–ΔS) = 37.8 J kg –1 K –1 at 5 K under a field variation (ΔH) of 7 T and its large estimated RCP of 623.6 J Kg –1 is comparable with those of typical MC materials. Both (–ΔS) and RCP are shown to scale with the saturation magnetization MS, suggesting that MS is the crucial factor controlling their magnitudes. Assuming (–ΔS) ~ (ΔH) n , the temperature dependence of n for the six samples are determined, n varying between 1.3 at 5 K to n = 2.2 at 130 K in line with its expected magnitudes based on mean-field theory. Furthermore, these results on structure-property correlations and scaling in GdCrO 3 suggest that its MC properties are tunable for potential low-temperature magnetic refrigeration applications.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Photoluminescence, photophysics, and photochemistry of the V B - defect in hexagonal boron nitride

Extensive photochemical and spectroscopic properties of the V B - defect in hexagonal boron nitride are calculated, concluding that the observed photoemission associated with recently observed optically detected magnetic resonance is most likely of (1) 3 E'' → (1) 3 A' 2 origin. Rapid intersystem crossing from the defect's triplet to singlet manifolds explains the observed short excited-state lifetime and very low quantum yield. New experimental results reveal smaller intrinsic spectral bandwidths than previously recognized, interpreted in terms of spectral narrowing and zero-phonon-line shifting induced by the Jahn-Teller effect. Different types of computational methods are applied to map out the complex triplet and singlet defect manifolds, including the doubly ionized formulation of the equation-of-motion coupled-cluster theory that is designed to deal with the open-shell nature of defect states, and mixed quantum-mechanics/molecular-mechanics schemes enabling 5763-atom simulations. Finally, two other energetically feasible spectral assignments from among the singlet and triplet manifolds are considered, but ruled out based on inappropriate photochemical properties.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Tailoring physical properties of crystals through synthetic temperature control: A case study for new polymorphic NbFeT e 2 phases

Growth parameters play a significant role in the crystal quality and physical properties of layered materials. Here we present a case study on a van der Waals magnetic NbFeTe 2 material. Two different types of polymorphic NbFeTe 2 phases, synthesized at different temperatures, display significantly different behaviors in crystal symmetry, electronic structure, electrical transport, and magnetism. While the phase synthesized at low temperature showing behavior consistent with previous reports, the new phase synthesized at high temperature, has completely different physical properties, such as metallic resistivity, long-range ferromagnetic order, anomalous Hall effect, negative magnetoresistance, and distinct electronic structures. Neutron diffraction reveals out-of-plane ferromagnetism below 70 K, consistent with the electrical transport and magnetic susceptibility studies. Finally, our work suggests that simply tuning synthetic parameters in a controlled manner could be an effective route to alter the physical properties of existing materials potentially unlocking new states of matter, or even discovering new materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Photoinduced Dirac semimetal in ZrTe5

Abstract Novel phases of matter with unique properties that emerge from quantum and topological protection present an important thrust of modern research. Of particular interest is to engineer these phases on demand using ultrafast external stimuli, such as photoexcitation, which offers prospects of their integration into future devices compatible with optical communication and information technology. Here, we use MeV Ultrafast Electron Diffraction (UED) to show how a transient three-dimensional (3D) Dirac semimetal state can be induced by a femtosecond laser pulse in a topological insulator ZrTe 5 . We observe marked changes in Bragg diffraction, which are characteristic of bond distortions in the photoinduced state. Using the atomic positions refined from the UED, we perform density functional theory (DFT) analysis of the electronic band structure. Our results reveal that the equilibrium state of ZrTe 5 is a topological insulator with a small band gap of ~ 25 meV, consistent with angle-resolved photoemission (ARPES) experiments. However, the gap is closed in the presence of strong spin-orbit coupling (SOC) in the photoinduced transient state, where massless Dirac fermions emerge in the chiral band structure. The time scale of the relaxation dynamics to the transient Dirac semimetal state is remarkably long, τ ~ 160 ps, which is two orders of magnitude longer than the conventional phonon-driven structural relaxation. The long relaxation is consistent with the vanishing density of states in Dirac spectrum and slow spin-repolarization of the SOC-controlled band structure accompanying the emergence of Dirac fermions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Multiphonon diffuse scattering in solids from first principles: Application to layered crystals and two-dimensional materials

Time-resolved diffuse scattering experiments have gained increasing attention due to their potential to reveal nonequilibrium dynamics of crystal lattice vibrations with full momentum resolution. Although progress has been made in interpreting experimental data on the basis of one-phonon scattering, understanding the role of individual phonons can be sometimes hindered by multiphonon excitations. In Ref. [M. Zacharias, H. Seiler, F. Caruso, D. Zahn, F. Giustino, P. C. Kelires, and R. Ernstorfer, Phys. Rev. Lett. 127, 207401 (2021)], we have introduced a rigorous approach for the calculation of the all-phonon inelastic scattering intensity of solids from first-principles. In the present work, we describe our implementation in detail and show that multiphonon interactions are captured efficiently by exploiting translational and time-reversal symmetries of the crystal. We demonstrate its predictive power by calculating the scattering patterns of monolayer molybdenum disulfide (MoS 2 ), bulk MoS 2 , and black phosphorus (bP), and we obtain excellent agreement with our measurements of thermal electron diffuse scattering. Remarkably, our results show that multiphonon excitations dominate in bP across multiple Brillouin zones, while in MoS 2 they play a less pronounced role. We expand our analysis for each system and examine the effect of individual atomic and interatomic vibrational motion on the diffuse scattering signals. We further demonstrate the high-throughput capability of our approach by reporting all-phonon scattering maps of two-dimensional MoSe 2 , WSe 2 , WS 2 , graphene, and CdI 2 , rationalizing in each case the effect of multiphonon processes. As a side point, we show that the special displacement method reproduces the thermally distorted configuration that generates precisely the all-phonon diffuse pattern. In conclusion, the present methodology opens the way for systematic calculations of the scattering intensity in crystals and the accurate interpretation of static and time-resolved inelastic scattering experiments.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Universality of Collective Density Fluctuations in Liquids at and away from Equilibrium – An Integrated Neutron Scattering, Theoretical, and Computational Study (Final Technical Report)

Liquids, ubiquitous on earth, are prototypical disordered condensed matter. As the mediate phase between non-structured gases and ordered solids, liquids p ossess intriguing complexities that are absent on either extreme counter phase, especially when they are driven away from equilibrium. Despite decades of studies, the physics of liquids still keeps surprising us and challenges our understanding of condensed matter, emergent phenomena, and complex systems. Among the many interesting properties, the nature of collective dynamics in liquids remains elusive because of both the presence of strong interactions and the absence of translational invariance. This project studied the universal features of collective density fluctuations (or collective modes), which include both phonon-like excitations and collective relaxations, in the Q-dependent generalized hydrodynamic regime in liquids and liquid-like matter both at and away from equilibrium. We performed systematic studies of the fast collective dynamics (including both phonon-like excitations and collective relaxations) of four classes of liquids characterized by different fragilities and interactions using synergistically integrated coherent neutron scattering experiments (Inelastic Neutron Scattering (INS), Quasi-Elastic Neutron Scattering (QENS), and Neutron Spin Echo (NSE)), a ViscoElastic Hydrodynamics theory we have been developing, andMolecular Dynamics (MD) simulations and our analysis package LiquidLib. The transformative knowledge of the collective dynamics in liquids is important to understand and ultimately control the transport of mass, energy, and charge in liquids, both at and away from thermal equilibrium, which may will likely impact a wide range of disciplines.

36 MATERIALS SCIENCE↗

Hund’s coupling mediated multi-channel quantum phase transition of a single magnetic impurity in Fe(Se, Te)

Understanding the interplay between individual magnetic impurities and superconductivity is crucial for bottom-up construction of novel phases of matter. Sub-gap bound states that are used in this endeavor are typically considered as independent entities that each result from the exchange scattering between the respective impurity orbitals and electrons of the superconducting condensate. Here we present experimental evidence of individual multi-spin impurities where the sub-gap states are not independent. Specifically, we find that by tuning the energy of the state closest to zero through zero, all other sub-gap states change particle-hole asymmetry as well. We show that this can be understood by including Hund’s coupling, which favors high-spin configurations, into a multi-orbital Anderson model. Unlike for the case of independent spins, the transition we observe signals the simultaneous departure of more than one quasiparticle from the impurity, while the parity of the ground state may remain unchanged. Our results show that Hund’s coupling is not only crucial in generating high-spin impurities, but also to understand the transition between two distinct ground states, and should therefore be taken into account for e.g. impurity-based band-structure engineering.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Unraveling the structural dependency of Weyl nodes in Co 2 MnGa

Conventionally, the modulation of the intrinsic Weyl nodes in Weyl semimetals is challenging, due to topological protection. Here we report the structural dependence of the Weyl nodes in a Co2⁢MnGa Heusler thin film via a temperature-dependent tetragonal distortion. The ability to manipulate these Weyl nodes allows for the control of the intrinsic electromagnetic properties. Temperature-dependent x-ray diffraction (XRD) measurements identify a compressive tetragonal distortion with decreasing temperature from 300 to 20 K. The calculated Weyl properties can be directly compared with experimental parameters through the temperature-dependent XRD measurements which show the intrinsic correlation between Weyl properties and important magnetic parameters. Further, the microscopic momentum space properties of Weyl nodes such as the distance (d W ), solid angle (Ω W ), tilt (φ W ), and nodal point energy (E W ) directly affect the macroscopic observable properties such as exchange stiffness (A), magnetization (M), and effective anisotropy field $H^{eff}_{K}$, as shown via structure-dependent density functional theory calculations. These predictions are experimentally observed as large variations in the bulk magnetization and effective anisotropy field as a function of temperature. These results highlight a unique degree of freedom in the control of macroscopic magnetic properties via the modulation of the intrinsic properties of Weyl nodes through structural distortions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Matter-wave stripe solitons induced by helicoidal spin–orbit coupling

Highlights: • Four kinds of exact stripe soliton solutions in Bose–Einstein condensates with helicoidal spin–orbit coupling (SOC) are represented. • The influence of helicoidal SOC on the structure of stripe solitons are described in detail. • The corresponding spin polarization is expressed on the Bloch sphere. • The formation mechanism of stripe solitons and the interaction properties of moving stripe solitons are discussed. We present the exact analytical solution of stripe solitons on different backgrounds in Bose–Einstein condensates with helicoidal spin–orbit coupling (SOC). They are divided into two categories, one is stationary stripe soliton on zero background or period-wave background, the other is moving stripe soliton on zero background or plane-wave background. The period of stripe is determined by both strength and frequency of helicoidal SOC, and the oscillation amplitude of stripe is adjustable in the case of fixed total density. The corresponding spin polarization is expressed on Bloch sphere. For stationary soliton, the spin density vector take values in three directions, which is different from the Rashba–Dresslhaus coupling scheme. The spin density vectors display more complicated spatial distribution for moving stripe solitons. Meanwhile, the formation mechanism of stripe soliton and the interaction properties of moving stripe solitons are discussed. Our work enhances the understanding of stripe solitons, and is helpful for the observation of stripe phase.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Disorder, interactions, and their interplay in novel narrow-gap Dirac materials and Weyl semimetals

Progress of the modern day condensed matter physics is to a large extent driven by the synthesis of new materials, advances in their experimental characterization and theoretical description. Recent discoveries of novel gapless Weyl semimetals, such as NaBi,CdAs, and BiTe-based films, in which magnetic dopants essentially suppress the gap, have added to the family of graphene and topological insulators actively investigated over the past decade. With the field of novel semimetals rapidly maturing, its focus necessarily shifts from demonstrations of the feasibility of such materials to their quantitative characterization. While the transport and optical properties of graphene and topological insulators are well captured within the picture of free non-interacting electrons, gapless 3D Weyl semimetals and narrow-gap 2D semiconductors with Dirac spectrum are known to be extremely susceptible to disorder and electron-electron interactions. This susceptibility obscures the manifestations of nontrivial band structure -- like quantum anomalous Hall effect -- of the new topological materials. Among particular projects to be addressed are: 1) optical conductivity of 3D gapless Dirac fermions in the presence of smooth disorder, 2) interplay of disorder and Coulomb interactions in the spectral properties of such fermions, 3) formation and structure of the impurity band with Coulomb supercritical clusters, 4) Coulomb interaction-driven renormalization of the electron spectrum and of the transport response in the presence of strong magnetic field, 5) instanton approach to the disorder-induced fluctuation states in zero-gap 3D materials, and 6) the role of disorder in quantum anomalous Hall effect. The proposal relies upon the investigators' previous broad expertise in interacting and disordered electron systems. The methods to be employed include perturbative diagrammatic technique, non-perturbative instanton and self-consistent approximations, hydrodynamics of electron liquid. Both analytical as well as numerical approaches are to be employed. The anticipated broader outcome of the proposal includes gaining an in-depth understanding of the interplay of the disorder and interactions under the conditions when this interplay has the most dramatic impact on observables. Traditionally, interaction effects are among the most challenging and interesting problems of condensed matter physics. Similarly, disordered systems typically present very difficult but extremely rich problems in the description of various materials. Importantly, understanding the spectral and transport properties of such materials not only presents the fundamental objective, but is also of particular interest for many applications, such as computation, memory, optics, plasmonics. In particular realization of the quantum anomalous Hall effect may lead to the development of low-power-consumption electronics. Indeed, a major constraint for practical use of the quantum Hall effect is limited by the requirement of the quantizing magnetic field. At the same time, the quantum anomalous Hall effect samples exhibit non-dissipative edge quantum transport in a zero magnetic field.

36 MATERIALS SCIENCE↗

Absence of bulk charge density wave order in the normal state of UTe 2

A spatially modulated superconducting state, known as pair density wave (PDW), is a tantalizing state of matter with unique properties. Recent scanning tunneling microscopy (STM) studies revealed that spin-triplet superconductor UTe 2 hosts an unprecedented spin-triplet, multi-component PDW whose three wavevectors are indistinguishable from a preceding charge-density wave (CDW) order that survives to temperatures well above the superconducting critical temperature, T c . Whether the PDW is the mother or a subordinate order remains unsettled. Here, based on a systematic search for bulk charge order above T c using resonant elastic X-ray scattering (REXS), we show that the structure factor of charge order previously identified by STM is absent in the bulk within the sensitivity of REXS. Our results invite two scenarios: either the density-wave orders condense simultaneously at T c in the bulk, in which case PDW order is likely the mother phase, or the charge modulations are restricted to the surface.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗