Extended phonon collapse and the origin of the charge-density wave in 2H-NbSe2

Intercalating Architecture for the Design of Charge Density Wave in Metallic MA2Z4 Materials

We present a novel approach to induce charge density waves (CDWs) in metallic MA2Z4 materials, resembling the behavior observed in transition metal dichalcogenides (TMDCs). This method leverages the intercalating architecture to maintain the same crystal field and Fermi surface topologies. Our investigation reveals that CDW instability in these materials arises from electron-phonon coupling (EPC) between the d band and longitudinal acoustic (LA) phonons, mirroring TMDC's behavior. By combining α-MA2Z4 with 1H-MX2 materials in a predictive CDW phase diagram using critical EPC constants, we demonstrate the feasibility of extending CDW across material families with comparable crystal fields and reveal the crucial role in CDW instability of the competition between ionic charge transfer and electron correlation. We further uncover a strain-induced Mott transition in β2-NbGe2N4 monolayer featuring star-of-David patterns. This work highlights the potential of intercalating architecture to engineer CDW materials, expanding our understanding of CDW instability and correlation physics.

Incognizant 1T/1H Charge-Density-Wave Phases in Monolayer NbTe2

While experimental realization of multiple charge-density waves (CDWs) has been ascribed to monolayer 1T-NbTe2, their atomic structures are still largely unclear, preventing a deep understanding of their novel electronic structures. Here, comparing first-principles-calculated orbital textures with reported STM measurements, we successfully identify multiple CDWs in monolayer NbTe2. Surprisingly, we reveal that both 1T/1H phases could exist in monolayer NbTe2, which was incognizant before. Particularly, we find that the experimentally observed 4 × 1 and 4 × 4 CDWs could be attributed to 1H stacking, while the observed 19 × 19 phase could possess 1T stacking. The existence of 1T/1H phases results in competition between CDW, spin-density wave (SDW), and ferromagnetism in 1H stacking under an external field and results in CDW-induced quantum phase transitions from a Kramers-Weyl fermion to a topological insulator in 1T stacking. Our study suggests NbTe2 as an exotic platform to investigate the interplay between CDW, SDW, and topological phases, which are largely unexplored in current experiments.

Self-Stacked 1T-1H Layers in 6R-NbSeTe and the Emergence of Charge and Magnetic Correlations Due to Ligand Disorder

The emergence of correlated phenomena arising from the combination of 1T and 1H van der Waals layers is the focus of intense research. Here, we synthesize a self-stacked 6R phase in NbSeTe, showing perfect alternating 1T and 1H layers that grow coherently along the c-direction, as revealed by scanning transmission electron microscopy. Angle-resolved photoemission spectroscopy shows a mixed contribution of the trigonal and octahedral Nb bands to the Fermi level. Diffuse scattering reveals temperature-independent short-range charge fluctuations with propagation vector qCO = (0.25 0), derived from the condensation of a longitudinal mode in the 1T layer, while the long-range charge density wave is quenched by ligand disorder. Magnetization measurements suggest the presence of an inhomogeneous, short-range magnetic order, further supported by the absence of a clear phase transition in the specific heat. These experimental analyses in combination with ab initio calculations indicate that the ground state of 6R-NbSeTe is described by a statistical distribution of short-range charge-modulated and spin-correlated regions driven by ligand disorder. Our results demonstrate how natural 1T-1H self-stacked bulk heterostructures can be used to engineer emergent phases of matter.

Exceptionally Slow, Long-Range, and Non-Gaussian Critical Fluctuations Dominate the Charge Density Wave Transition

(TaSe_{4})_{2}I is a well-studied quasi-one-dimensional compound long-known to have a charge-density wave (CDW) transition around 263 K. We argue that the critical fluctuations of the pinned CDW order parameter near the transition can be inferred from the resistance noise on account of their coupling to the dissipative normal carriers. Remarkably, the critical fluctuations of the CDW order parameter are slow enough to survive the thermodynamic limit and dominate the low-frequency resistance noise. The noise variance and relaxation time show rapid growth (critical opalescence and critical slowing down) within a temperature window of ϵ≈±0.1, where ϵ is the reduced temperature. This is very wide but consistent with the Ginzburg criterion. We further show that this resistance noise can be quantitatively used to extract the associated critical exponents. Below |ϵ|≲0.02, we observe a crossover from mean-field to a fluctuation-dominated regime with the critical exponents taking anomalously low values. The distribution of fluctuations in the critical transition region is skewed and strongly non-Gaussian. This non-Gaussianity is interpreted as the breakdown of the validity of the central limit theorem as the diverging coherence volume becomes comparable to the macroscopic sample size. The large magnitude critical fluctuations observed over an extended temperature range, as well as the crossover from the mean-field to the fluctuation-dominated regime highlight the role of the quasi-one-dimensional character in controlling the phase transition.

Charge density waves in two-dimensional transition metal dichalcogenides

Charge density wave (CDW is one of the most ubiquitous electronic orders in quantum materials. While the essential ingredients of CDW order have been extensively studied, a comprehensive microscopic understanding is yet to be reached. Recent research efforts on the CDW phenomena in two-dimensional (2D) materials provide a new pathway toward a deeper understanding of its complexity. This review provides an overview of the CDW orders in 2D with atomically thin transition metal dichalcogenides (TMDCs) as the materials platform. We mainly focus on the electronic structure investigations on the epitaxially grown TMDC samples with angle-resolved photoemission spectroscopy and scanning tunneling microscopy/spectroscopy as complementary experimental tools. We discuss the possible origins of the 2D CDW, novel quantum states coexisting with them, and exotic types of charge orders that can only be realized in the 2D limit.

Unconventional Charge-Density-Wave Gap in Monolayer NbS2

Using scanning tunneling microscopy and spectroscopy, for a monolayer of transition metal dichalcogenide H-NbS2 grown by molecular beam epitaxy on graphene, we provide unambiguous evidence for a charge density wave (CDW) with a 3 × 3 superstructure, which is not present in bulk NbS2. Local spectroscopy displays a pronounced gap on the order of 20 meV at the Fermi level. Within the gap, low-energy features are present. The gap structure with its low-energy features is at variance with the expectation for a gap opening in the electronic band structure due to a CDW. Instead, comparison with ab initio calculations indicates that the observed gap structure must be attributed to combined electron-phonon quasiparticles. The phonons in question are the elusive amplitude and phase collective modes of the CDW transition. Our findings advance the understanding of CDW mechanisms in 2D materials and their spectroscopic signatures.

Structural approach to charge density waves in low-dimensional systems: electronic instability and chemical bonding

The charge density wave (CDW) instability, usually occurring in low-dimensional metals, has been a topic of interest for longtime. However, some very fundamental aspects of the mechanism remain unclear. Recently, a plethora of new CDW materials, a substantial fraction of which is two-dimensional or even three-dimensional, has been prepared and characterised as bulk and/or single-layers. As a result, the need for revisiting the primary mechanism of the instability, based on the electron-hole instability established more than 50 years ago for quasi-one-dimensional (quasi-1D) conductors, has clearly emerged. In this work, we consider a large number of CDW materials to revisit the main concepts used in understanding the CDW instability, and emphasise the key role of the momentum dependent electron-phonon coupling in linking electronic and structural degrees of freedom. We argue that for quasi-1D systems, earlier weak coupling theories work appropriately and the energy gain due to the CDW and the concomitant periodic lattice distortion (PLD) remains primarily due to a Fermi surface nesting mechanism. However, for materials with higher dimensionality, intermediate and strong coupling regimes are generally at work and the modification of the chemical bonding network by the PLD is at the heart of the instability. We emphasise the need for a microscopic approach blending condensed matter physics concepts and state-of-the-art first-principles calculations with quite fundamental chemical bonding ideas in understanding the CDW phenomenon in these materials.

Theoretical study of CDW phases for bulk NbX2 (X = S and Se)

In most two-dimensional transition metal chalcogenides, the superconducting phase coexists with the charge density wave (CDW) phase. There exists at least one case, i.e. bulk 2H-NbS2, that does not conform to this picture. Scientists have shown great interest in trying to experimentally find the CDW phase of bulk NbS2 since 1975. Is there any theoretically more stable thermodynamic state than its higher-temperature metal phase, especially in the case of charge injection? Theoretically more stable CDW bulk configurations (TC for 2H-NbS2 and TTs for 2H-NbSe2) with partial pseudo energy gaps were predicted through the harmonic phonon softening theory and first-principles calculations. The ratios of larger to smaller pseudo gaps around K-H segment in the Brillouin zone for CDW phases are basically equal to those of superconductivity phases for bulk 2H-NbX2 (X = S and Se). The CDW phase should coexist with its superconductor state below the critical temperature rather than the metal phase for bulk 2H-NbS2. The presence of CDW phase should be more easily observed experimentally when the injected charge reaches 0.5e/Nb18S36 for bulk 2H-NbS2. Our calculations of density of state (DOS) indicated that, during Nb atoms contracting to form the CDW phases with symmetry breaking in the in-plane direction, dominant conductive carriers are always of p-type for bulk 2H-NbS2 while the alternation of carrier type from p-type to n-type occurs for bulk 2H-NbSe2. The Fermi level continuously drops and then the M-L segment of the out-of-plane energy band emerges from the Fermi surface, which corresponds to the reversal of p-n type sign. Lifshitz transition of pocket-vanishing types occurs in the out-of-plane direction without symmetry breaking during the geometrical structural phase transition for bulk 2H-NbSe2. Our calculations have theoretically addressed the long-standing coexistence issue of CDW and superconducting phases.

Competing charge-density wave instabilities in the kagome metal ScV6Sn6

Owing to its unique geometry, the kagome lattice hosts various many-body quantum states including frustrated magnetism, superconductivity, and charge-density waves (CDWs). In this work, using inelastic X-ray scattering, we discover a dynamic short-range [Formula: see text] CDW that is dominant in the kagome metal ScV6Sn6 above TCDW ≈ 91 K, competing with the [Formula: see text] CDW that orders below TCDW. The competing CDW instabilities lead to an unusual CDW formation process, with the most pronounced phonon softening and the static CDW occurring at different wavevectors. First-principles calculations indicate that the [Formula: see text] CDW is energetically favored, while a wavevector-dependent electron-phonon coupling (EPC) promotes the [Formula: see text] CDW as the ground state, and leads to enhanced electron scattering above TCDW. These findings underscore EPC-driven correlated many-body physics in ScV6Sn6 and motivate studies of emergent quantum phases in the strong EPC regime.

Chiral charge density wave induced by mirror symmetry breaking in kagome metal

Kagome lattice provides a distinctive platform to investigate various correlated electron orders. Recently, an unconventional charge density wave (CDW) with novel chirality is observed in the kagome metalAV3Sb5(A= K, Rb, Cs), and the origin of which is still unclear. Here, using a tight-binding model and the mean-field method, we calculate the electron order in the quasi-two-dimensional kagome lattice with 1/3 electron filling, and show that the chiral CDW emerges under a set of parameters withC6rotational symmetry but without mirror symmetry. Physically, the reason why we choose this set of parameters is based on the possible tangential distortion of the kagome lattice. Our results provide a fresh insight to understand the microscopic origin of the unconventional CDW inAV3Sb5.

Precursor region with full phonon softening above the charge-density-wave phase transition in 2H-TaSe2

Research on charge-density-wave (CDW) ordered transition-metal dichalcogenides continues to unravel new states of quantum matter correlated to the intertwined lattice and electronic degrees of freedom. Here, we report an inelastic x-ray scattering investigation of the lattice dynamics of the canonical CDW compound 2H-TaSe2 complemented by angle-resolved photoemission spectroscopy and density functional perturbation theory. Our results rule out the formation of a central-peak without full phonon softening for the CDW transition in 2H-TaSe2 and provide evidence for a novel precursor region above the CDW transition temperature TCDW, which is characterized by an overdamped phonon mode and not detectable in our photoemission experiments. Thus, 2H-TaSe2 exhibits structural before electronic static order and emphasizes the important lattice contribution to CDW transitions. Our ab-initio calculations explain the interplay of electron-phonon coupling and Fermi surface topology triggering the CDW phase transition and predict that the CDW soft phonon mode promotes emergent superconductivity near the pressure-driven CDW quantum critical point.

Discovery of Charge Order in the Transition Metal Dichalcogenide Fe_{x}NbS_{2}

The Fe intercalated transition metal dichalcogenide (TMD), Fe_{1/3}NbS_{2}, exhibits remarkable resistance switching properties and highly tunable spin ordering phases due to magnetic defects. We conduct synchrotron x-ray scattering measurements on both underintercalated (x=0.32) and overintercalated (x=0.35) samples. We discover a new charge order phase in the overintercalated sample, where the excess Fe atoms lead to a zigzag antiferromagnetic order. The agreement between the charge and magnetic ordering temperatures, as well as their intensity relationship, suggests a strong magnetoelastic coupling as the mechanism for the charge ordering. Our results reveal the first example of a charge order phase among the intercalated TMD family and demonstrate the ability to stabilize charge modulation by introducing electronic correlations, where the charge order is absent in bulk 2H-NbS_{2} compared to other pristine TMDs.

Softening of a flat phonon mode in the kagome ScV6Sn6

Geometrically frustrated kagome lattices are raising as novel platforms to engineer correlated topological electron flat bands that are prominent to electronic instabilities. Here, we demonstrate a phonon softening at the kz = π plane in ScV6Sn6. The low energy longitudinal phonon collapses at ~98 K and q = [Formula: see text] due to the electron-phonon interaction, without the emergence of long-range charge order which sets in at a different propagation vector qCDW = [Formula: see text]. Theoretical calculations corroborate the experimental finding to indicate that the leading instability is located at [Formula: see text] of a rather flat mode. We relate the phonon renormalization to the orbital-resolved susceptibility of the trigonal Sn atoms and explain the approximately flat phonon dispersion. Our data report the first example of the collapse of a kagome bosonic mode and promote the 166 compounds of kagomes as primary candidates to explore correlated flat phonon-topological flat electron physics.

Signature of spin-phonon coupling driven charge density wave in a kagome magnet

The intertwining between spin, charge, and lattice degrees of freedom can give rise to unusual macroscopic quantum states, including high-temperature superconductivity and quantum anomalous Hall effects. Recently, a charge density wave (CDW) has been observed in the kagome antiferromagnet FeGe, indicative of possible intertwining physics. An outstanding question is that whether magnetic correlation is fundamental for the spontaneous spatial symmetry breaking orders. Here, utilizing elastic and high-resolution inelastic x-ray scattering, we observe a c-axis superlattice vector that coexists with the 2[Formula: see text]2[Formula: see text]1 CDW vectors in the kagome plane. Most interestingly, between the magnetic and CDW transition temperatures, the phonon dynamical structure factor shows a giant phonon-energy hardening and a substantial phonon linewidth broadening near the c-axis wavevectors, both signaling the spin-phonon coupling. By first principles and model calculations, we show that both the static spin polarization and dynamic spin excitations intertwine with the phonon to drive the spatial symmetry breaking in FeGe.

An electronic origin of charge order in infinite-layer nickelates

A charge order (CO) with a wavevector [Formula: see text] is observed in infinite-layer nickelates. Here we use first-principles calculations to demonstrate a charge-transfer-driven CO mechanism in infinite-layer nickelates, which leads to a characteristic Ni1+-Ni2+-Ni1+ stripe state. For every three Ni atoms, due to the presence of near-Fermi-level conduction bands, Hubbard interaction on Ni-d orbitals transfers electrons on one Ni atom to conduction bands and leaves electrons on the other two Ni atoms to become more localized. We further derive a low-energy effective model to elucidate that the CO state arises from a delicate competition between Hubbard interaction on Ni-d orbitals and charge transfer energy between Ni-d orbitals and conduction bands. With physically reasonable parameters, [Formula: see text] CO state is more stable than uniform paramagnetic state and usual checkerboard antiferromagnetic state. Our work highlights the multi-band nature of infinite-layer nickelates, which leads to some distinctive correlated properties that are not found in cuprates.

Influence of the physical properties of the layered oxyselenides Bi2YO4Cu2Se2 through Ni doping

We have synthesized a series of Ni-doped layered oxyselenides Bi2YO4Cu2-xNixSe2 (0 ≤ x ≤ 0.4). The crystal structure and physical properties were studied through X-ray diffraction, and electric and thermo transport measurements. We also performed DFT calculations to study the electric structure of the designed Bi2YO4Ni2Se2, which is similar to that of KNi2Se2.

Intertwining of Magnetism and Charge Ordering in Kagome FeGe

Recent experiments report a charge density wave (CDW) in the antiferromagnet FeGe, but the nature of the charge ordering and the associated structural distortion remains elusive. We discuss the structural and electronic properties of FeGe. Our proposed ground state phase accurately captures atomic topographies acquired by scanning tunneling microscopy. We show that the 2 × 2 × 1 CDW likely results from the Fermi surface nesting of hexagonal-prism-shaped kagome states. FeGe is found to exhibit distortions in the positions of the Ge atoms instead of the Fe atoms in the kagome layers. Using in-depth first-principles calculations and analytical modeling, we demonstrate that this unconventional distortion is driven by the intertwining of magnetic exchange coupling and CDW interactions in this kagome material. The movement of Ge atoms from their pristine positions also enhances the magnetic moment of the Fe kagome layers. Our study indicates that magnetic kagome lattices provide a material candidate for exploring the effects of strong electronic correlations on the ground state and their implications for transport, magnetic, and optical responses in materials.

Possible enhancement of the superconductingTcdue to sharp Kohn-like soft phonon anomalies

Phonon softening is a ubiquitous phenomenon in condensed matter systems which is often associated with charge density wave (CDW) instabilities and anharmonicity. The interplay between phonon softening, CDW and superconductivity is a topic of intense debate. In this work, the effects of anomalous soft phonon instabilities on superconductivity are studied based on a recently developed theoretical framework that accounts for phonon damping and softening within the Migdal-Eliashberg theory. Model calculations show that the phonon softening in the form of a sharp dip in the phonon dispersion relation, either acoustic or optical (including the case of Kohn-type anomalies typically associated with CDW), can cause a manifold increase of the electron-phonon coupling constantλ. This, under certain conditions, which are consistent with the concept of optimal frequency introduced by Bergmann and Rainer, can produce a large increase of the superconducting transition temperatureTc. In summary, our results suggest the possibility of reaching high-temperature superconductivity by exploiting soft phonon anomalies restricted in momentum space.

Superconductivity and density-wave fluctuations in an extended triangular Hubbard model: an application to SnSe2

We employ the fluctuation-exchange approximation to study the relation of superconducting pairing symmetries and density-wave fluctuations based on the extended triangular Hubbard model upon electron doping and interactions, with an possible application to the layered metal dichalcogenide SnSe₂. For the case where the interactions between electrons contain only the on-site Hubbard term, the superconducting pairings are mainly mediated by spin fluctuations, and the spin-singlet pairing with thed-wave symmetry robustly dominates in the low and moderate doping levels, and ad-wave to extendeds-wave transition is observed as the electron doping reachesn = 1. When the near-neighbor site Coulomb interactions are also included, the charge fluctuations are enhanced, and the spin-triplet pairings with thep-wave andf-wave symmetries can be realized in the high and low doping levels, respectively.

Soft-Phonon and Charge-Density-Wave Formation in Nematic BaNi_{2}As_{2}

We use diffuse and inelastic x-ray scattering to study the formation of an incommensurate charge-density-wave (I-CDW) in BaNi_{2}As_{2}, a candidate system for charge-driven electronic nematicity. Intense diffuse scattering is observed around the modulation vector of the I-CDW, Q_{I-CDW}. It is already visible at room temperature and collapses into superstructure reflections in the long-range ordered state where a small orthorhombic distortion occurs. A clear dip in the dispersion of a low-energy transverse optical phonon mode is observed around Q_{I-CDW}. The phonon continuously softens upon cooling, ultimately driving the transition to the I-CDW state. The transverse character of the soft-phonon branch elucidates the complex pattern of the I-CDW satellites observed in the current and earlier studies and settles the debated unidirectional nature of the I-CDW. The phonon instability and its reciprocal space position are well captured by our ab initio calculations. These, however, indicate that neither Fermi surface nesting, nor enhanced momentum-dependent electron-phonon coupling can account for the I-CDW formation, demonstrating its unconventional nature.

Observation of anomalous amplitude modes in the kagome metal CsV3Sb5

The kagome lattice provides a fertile platform to explore novel symmetry-breaking states. Charge-density wave (CDW) instabilities have been recently discovered in a new kagome metal family, commonly considered to arise from Fermi-surface instabilities. Here we report the observation of Raman-active CDW amplitude modes in CsV₃Sb₅, which are collective excitations typically thought to emerge out of frozen soft phonons, although phonon softening is elusive experimentally. The amplitude modes strongly hybridize with other superlattice modes, imparting them with clear temperature-dependent frequency shift and broadening, rarely seen in other known CDW materials. Both the mode mixing and the large amplitude mode frequencies suggest that the CDW exhibits the character of strong electron-phonon coupling, a regime in which phonon softening can cease to exist. Our work highlights the importance of the lattice degree of freedom in the CDW formation and points to the complex nature of the mechanism.

The mechanism of the charge density wave in kagome metals is not fully understood. Here, the authors report the observation of unusual large-frequency collective lattice excitations, or amplitude modes, in CsV₃Sb₅ in the absence of phonon mode softening, evidencing the strong electron-phonon coupling regime.

Ubiquitous defect-induced density wave instability in monolayer graphene

Quantum materials are notoriously sensitive to their environments, where small perturbations can tip a system toward one of several competing ground states. Graphene hosts a rich assortment of such competing phases, including a bond density wave instability ("Kekulé distortion") that couples electrons at the K/K' valleys and breaks the lattice symmetry. Here, we report observations of a ubiquitous Kekulé distortion across multiple graphene systems. We show that extremely dilute concentrations of surface atoms (less than three adsorbed atoms every 1000 graphene unit cells) can self-assemble and trigger the onset of a global Kekulé density wave phase. Combining complementary momentum-sensitive angle-resolved photoemission spectroscopy (ARPES) and low-energy electron diffraction (LEED) measurements, we confirm the presence of this density wave phase and observe the opening of an energy gap. Our results reveal an unexpected sensitivity of the graphene lattice to dilute surface disorder and show that adsorbed atoms offer an attractive route toward designing novel phases in two-dimensional materials.

Nuclear Quantum Effects on the Charge-Density Wave Transition in NbX2 (X = S, Se)

Understanding the origin of charge-density wave (CDW) instability is important for manipulating novel collective electronic states. Many layered transition metal dichalcogenides (TMDs) share similarity in the structural and electronic instability, giving rise to diverse CDW phases and superconductivity. It is still puzzling that even isostructural and isoelectronic TMDs show distinct CDW features. For instance, bulk NbSe₂ exhibits CDW order at low temperature, while bulk NbS₂ displays no CDW instability. The CDW transitions in single-layer NbS₂ and NbSe₂ are also different. In the classic limit, we investigate the electron correlation effects on the dimensionality dependence of the CDW ordering. By performing ab initio path integral molecular dynamics simulations and comparative analyses, we further revealed significant nuclear quantum effects in these systems. Specifically, the quantum motion of sulfur anions significantly reduces the CDW transition temperature in both bulk and single-layer NbS₂, resulting in distinct CDW features in the NbS₂ and NbSe₂ systems.

Confinement-Engineered Superconductor to Correlated-Insulator Transition in a van der Waals Monolayer

Transition metal dichalcogenides (TMDC) are a rich family of two-dimensional materials displaying a multitude of different quantum ground states. In particular, d3 TMDCs are paradigmatic materials hosting a variety of symmetry broken states, including charge density waves, superconductivity, and magnetism. Among this family, NbSe2 is one of the best-studied superconducting materials down to the monolayer limit. Despite its superconducting nature, a variety of results point toward strong electronic repulsions in NbSe2. Here, we control the strength of the interactions experimentally via quantum confinement and use low-temperature scanning tunneling microscopy (STM) and spectroscopy (STS) to demonstrate that NbSe2 is in close proximity to a correlated insulating state. This reveals the coexistence of competing interactions in NbSe2, creating a transition from a superconducting to an insulating quantum correlated state by confinement-controlled interactions. Our results demonstrate the dramatic role of interactions in NbSe2, establishing NbSe2 as a correlated superconductor with competing interactions.

Computational Methods for Charge Density Waves in 2D Materials

Two-dimensional (2D) materials that exhibit charge density waves (CDWs)-spontaneous reorganization of their electrons into a periodic modulation-have generated many research endeavors in the hopes of employing their exotic properties for various quantum-based technologies. Early investigations surrounding CDWs were mostly focused on bulk materials. However, applications for quantum devices require few-layer materials to fully utilize the emergent phenomena. The CDW field has greatly expanded over the decades, warranting a focus on the computational efforts surrounding them specifically in 2D materials. In this review, we cover ground in the following relevant theory-driven subtopics for TaS2 and TaSe2: summary of general computational techniques and methods, resulting atomic structures, the effect of electron-phonon interaction of the Raman scattering modes, the effects of confinement and dimensionality on the CDW, and we end with a future outlook. Through understanding how the computational methods have enabled incredible advancements in quantum materials, one may anticipate the ever-expanding directions available for continued pursuit as the field brings us through the 21st century.

Electron-momentum dependence of electron-phonon coupling underlies dramatic phonon renormalization in YNi2B2C

Electron-phonon coupling, i.e., the scattering of lattice vibrations by electrons and vice versa, is ubiquitous in solids and can lead to emergent ground states such as superconductivity and charge-density wave order. A broad spectral phonon line shape is often interpreted as a marker of strong electron-phonon coupling associated with Fermi surface nesting, i.e., parallel sections of the Fermi surface connected by the phonon momentum. Alternatively broad phonons are known to arise from strong atomic lattice anharmonicity. Here, we show that strong phonon broadening can occur in the absence of both Fermi surface nesting and lattice anharmonicity, if electron-phonon coupling is strongly enhanced for specific values of electron-momentum, k. We use inelastic neutron scattering, soft x-ray angle-resolved photoemission spectroscopy measurements and ab-initio lattice dynamical and electronic band structure calculations to demonstrate this scenario in the highly anisotropic tetragonal electron-phonon superconductor YNi2B2C. This new scenario likely applies to a wide range of compounds.

Dispersion of the surface phonons in semiconductor/topological insulator Si/Bi2Te3 heterostructure studied by high resolution Brillouin spectroscopy

The dynamics and dispersion of surface phonons in heterostructure semiconductor/ topological insulator Si/Bi₂Te₃ was investigated using high resolution Brillouin light scattering method in the GHz frequency range. Both Rayleigh and Sezawa surface acoustic waves have been observed for wave vectors ranging from 0.006 to 0.023 nm^-1. Anomaly in dispersion relations ω(q) for both surface waves were detected for the wave vector q = 0.016 nm^-1. The finite element method (FEM) was used to simulate the observed shapes of ω(q) and to find the deformation profiles of surface acoustic waves. We attribute the observed changes to the coupling between low energy electrons and surface phonons. The coupling between helical Dirac states and surface phonons is discussed in the frame of accessible theoretical models.

A full gap above the Fermi level: the charge density wave of monolayer VS2

In the standard model of charge density wave (CDW) transitions, the displacement along a single phonon mode lowers the total electronic energy by creating a gap at the Fermi level, making the CDW a metal-insulator transition. Here, using scanning tunneling microscopy and spectroscopy and ab initio calculations, we show that VS2 realizes a CDW which stands out of this standard model. There is a full CDW gap residing in the unoccupied states of monolayer VS2. At the Fermi level, the CDW induces a topological metal-metal (Lifshitz) transition. Non-linear coupling of transverse and longitudinal phonons is essential for the formation of the CDW and the full gap above the Fermi level. Additionally, x-ray magnetic circular dichroism reveals the absence of net magnetization in this phase, pointing to coexisting charge and spin density waves in the ground state.

Role of Equilibrium Fluctuations in Light-Induced Order

Engineering novel states of matter with light is at the forefront of materials research. An intensely studied direction is to realize broken-symmetry phases that are "hidden" under equilibrium conditions but can be unleashed by an ultrashort laser pulse. Despite a plethora of experimental discoveries, the nature of these orders and how they transiently appear remain unclear. To this end, we investigate a nonequilibrium charge density wave (CDW) in rare-earth tritellurides, which is suppressed in equilibrium but emerges after photoexcitation. Using a pump-pump-probe protocol implemented in ultrafast electron diffraction, we demonstrate that the light-induced CDW consists solely of order parameter fluctuations, which bear striking similarities to critical fluctuations in equilibrium despite differences in the length scale. By calculating the dynamics of CDW fluctuations in a nonperturbative model, we further show that the strength of the light-induced order is governed by the amplitude of equilibrium fluctuations. These findings highlight photoinduced fluctuations as an important ingredient for the emergence of transient orders out of equilibrium. Our results further suggest that materials with strong fluctuations in equilibrium are promising platforms to host hidden orders after laser excitation.

The stochastic self-consistent harmonic approximation: calculating vibrational properties of materials with full quantum and anharmonic effects

The efficient and accurate calculation of how ionic quantum and thermal fluctuations impact the free energy of a crystal, its atomic structure, and phonon spectrum is one of the main challenges of solid state physics, especially when strong anharmonicy invalidates any perturbative approach. To tackle this problem, we present the implementation on a modular Python code of the stochastic self-consistent harmonic approximation (SSCHA) method. This technique rigorously describes the full thermodynamics of crystals accounting for nuclear quantum and thermal anharmonic fluctuations. The approach requires the evaluation of the Born-Oppenheimer energy, as well as its derivatives with respect to ionic positions (forces) and cell parameters (stress tensor) in supercells, which can be provided, for instance, by first principles density-functional-theory codes. The method performs crystal geometry relaxation on the quantum free energy landscape, optimizing the free energy with respect to all degrees of freedom of the crystal structure. It can be used to determine the phase diagram of any crystal at finite temperature. It enables the calculation of phase boundaries for both first-order and second-order phase transitions from the Hessian of the free energy. Finally, the code can also compute the anharmonic phonon spectra, including the phonon linewidths, as well as phonon spectral functions. We review the theoretical framework of the SSCHA and its dynamical extension, making particular emphasis on the physical inter pretation of the variables present in the theory that can enlighten the comparison with any other anharmonic theory. A modular and flexible Python environment is used for the implementation, which allows for a clean interaction with other packages. We briefly present a toy-model calculation to illustrate the potential of the code. Several applications of the method in superconducting hydrides, charge-density-wave materials, and thermoelectric compounds are also reviewed.

Charge order lock-in by electron-phonon coupling in La1.675Eu0.2Sr0.125CuO4

Charge order is universal to all hole-doped cuprates. Yet, the driving interactions remain an unsolved problem. Electron-electron interaction is widely believed to be essential, whereas the role of electron-phonon interaction is unclear. We report an ultrahigh-resolution resonant inelastic x-ray scattering (RIXS) study of the in-plane bond-stretching phonon mode in stripe-ordered cuprate La1.675Eu0.2Sr0.125CuO4 Phonon softening and lifetime shortening are found around the charge ordering wave vector. In addition to these self-energy effects, the electron-phonon coupling is probed by its proportionality to the RIXS cross section. We find an enhancement of the electron-phonon coupling around the charge-stripe ordering wave vector upon cooling into the low-temperature tetragonal structure phase. These results suggest that, in addition to electronic correlations, electron-phonon coupling contributes substantially to the emergence of long-range charge-stripe order in cuprates.

Superconductivity and strong anharmonicity in novel Nb-S phases

In this work we explore the phase diagram of the binary Nb-S system from ambient pressures up to 250 GPa usingab initioevolutionary crystal structure prediction. We find several new stable compositions and phases, especially in the high-pressure regime, and investigate their electronic, vibrational, and superconducting properties. Our calculations show that all materials, besides the low-pressure phases of pure sulfur, are metals with low electron-phonon (ep) coupling strengths and critical superconducting temperatures below 15 K. Furthermore, we investigate the effects of phonon anharmonicity on lattice dynamics, ep interactions, and superconductivity for the novel high-pressure phase of Nb₂S, demonstrating that the inclusion of anharmonicity stabilizes the lattice and enhances the ep interaction.

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.

van der Waals driven anharmonic melting of the 3D charge density wave in VSe2

Understanding of charge-density wave (CDW) phases is a main challenge in condensed matter due to their presence in high-Tc superconductors or transition metal dichalcogenides (TMDs). Among TMDs, the origin of the CDW in VSe₂ remains highly debated. Here, by means of inelastic x-ray scattering and first-principles calculations, we show that the CDW transition is driven by the collapse at 110 K of an acoustic mode at q_CDW = (2.25 0 0.7) r.l.u. The softening starts below 225 K and expands over a wide region of the Brillouin zone, identifying the electron-phonon interaction as the driving force of the CDW. This is supported by our calculations that determine a large momentum-dependence of the electron-phonon matrix-elements that peak at the CDW wave vector. Our first-principles anharmonic calculations reproduce the temperature dependence of the soft mode and the T_CDW onset only when considering the out-of-plane van der Waals interactions, which reveal crucial for the melting of the CDW phase.

The nature of the charge density wave transition in VSe₂ is still debated. Here, the authors demonstrate that the transition is mainly driven by electron-phonon interactions, despite the presence of the Fermi-surface nesting, and that Wan-der-Waals forces are responsible for melting of the charge density wave order.

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.

Chemical vapor deposition and temperature-dependent Raman characterization of two-dimensional vanadium ditelluride

Recently, ultrathin two-dimensional (2D) metallic vanadium dichalcogenides have attracted widespread attention because of the charge density wave (CDW) phase transition and possible ferromagnetism. Herein, we report the synthesis and temperature-dependent Raman characterization of the 2D vanadium ditelluride (VTe₂). The synthesis is done by atmospheric pressure chemical vapor deposition (APCVD) using vanadium chloride (VCl₃) precursor on fluorphlogopite mica, sapphire, and h-BN substrates. A large area of the thin film with thickness ∼10 nm is grown on the hexagonal boron nitride (h-BN) substrate. Temperature-dependent Raman characterization of VTe₂ is conducted from room temperature to 513 K. Remarkable changes of Raman modes at around 413 K are observed, indicating the structural phase transition.

Two-dimensional vanadium ditelluride has been synthesized on mica, sapphire, and h-BN substrates by atmospheric pressure chemical vapor deposition.

Examining Experimental Raman Mode Behavior in Mono- and Bilayer 2H-TaSe2 via Density Functional Theory: Implications for Quantum Information Science

Tantalum diselenide (TaSe2) is a metallic transition metal dichalcogenide whose structure and vibrational behavior strongly depend on temperature and thickness, and this behavior includes the emergence of charge density wave (CDW) states at very low temperatures. In this work, observed Raman modes for mono- and bilayer are described across several spectral regions and compared to those seen in the bulk case. These modes, which include an experimentally observed forbidden Raman mode and low-frequency CDWs, are then matched to corresponding vibrations predicted by density functional theory (DFT). The reported match between experimental and computational results supports the presented vibrational visualizations of these modes. Support is also provided by experimental phonons observed in additional Raman spectra as a function of temperature and thickness. These results highlight the importance of understanding CDWs since they are likely to play a fundamental role in the future realization of solid-state quantum information platforms based on nonequilibrium phenomena.

Measuring the Electron–Phonon Interaction in Two-Dimensional Superconductors with He-Atom Scattering

Helium-atom scattering (HAS) spectroscopy from conducting surfaces has been shown to provide direct information on the electron–phonon interaction, more specifically the mass-enhancement factor λ from the temperature dependence of the Debye–Waller exponent, and the mode-selected electron–phonon coupling constants λQν from the inelastic HAS intensities from individual surface phonons. The recent applications of the method to superconducting ultra-thin films, quasi-1D high-index surfaces, and layered transition-metal and topological pnictogen chalcogenides are briefly reviewed.

Visualizing the Anomalous Charge Density Wave States in Graphene/NbSe2 Heterostructures

Metallic layered transition metal dichalcogenides (TMDs) host collective many-body interactions, including the competing superconducting and charge density wave (CDW) states. Graphene is widely employed as a heteroepitaxial substrate for the growth of TMD layers and as an ohmic contact, where the graphene/TMD heterostructure is naturally formed. The presence of graphene can unpredictably influence the CDW order in 2D CDW conductors. This work reports the CDW transitions of 2H-NbSe₂ layers in graphene/NbSe₂ heterostructures. The evolution of Raman spectra demonstrates that the CDW phase transition temperatures (T_CDW ) of NbSe₂ are dramatically decreased when capped by graphene. The induced anomalous short-range CDW state is confirmed by scanning tunneling microscopy measurements. The findings propose a new criterion to determine the T_CDW through monitoring the line shape of the A_1g mode. Meanwhile, the 2D band is also discovered as an indicator to observe the CDW transitions. First-principles calculations imply that interfacial electron doping suppresses the CDW states by impeding the lattice distortion of 2H-NbSe₂ . The extraordinary random CDW lattice suggests deep insight into the formation mechanism of many collective electronic states and possesses great potential in modulating multifunctional devices.

Coexistence of Superconductivity and Charge Density Waves in Tantalum Disulfide: Experiment and Theory

The coexistence of charge density wave (CDW) and superconductivity in tantalum disulfide (2H-TaS_{2}) at low temperature is boosted by applying hydrostatic pressures to study both vibrational and magnetic transport properties. Around P_{c}, we observe a superconducting dome with a maximum superconducting transition temperature T_{c}=9.1  K. First-principles calculations of the electronic structure predict that, under ambient conditions, the undistorted structure is characterized by a phonon instability at finite momentum close to the experimental CDW wave vector. Upon compression, this instability is found to disappear, indicating the suppression of CDW order. The calculations reveal an electronic topological transition (ETT), which occurs before the suppression of the phonon instability, suggesting that the ETT alone is not directly causing the structural change in the system. The temperature dependence of the first vortex penetration field has been experimentally obtained by two independent methods. While a d wave and single-gap BCS prediction cannot describe the lower critical field H_{c1} data, the temperature dependence of the H_{c1} can be well described by a single-gap anisotropic s-wave order parameter.

Weak Dimensionality Dependence and Dominant Role of Ionic Fluctuations in the Charge-Density-Wave Transition of NbSe_{2}

Contradictory experiments have been reported about the dimensionality effect on the charge-density-wave transition in 2H NbSe_{2}. While scanning tunneling experiments on single layers grown by molecular beam epitaxy measure a charge-density-wave transition temperature in the monolayer similar to the bulk, around 33 K, Raman experiments on exfoliated samples observe a large enhancement of the transition temperature up to 145 K. By employing a nonperturbative approach to deal with anharmonicity, we calculate from first principles the temperature dependence of the phonon spectra both for bulk and monolayer. In both cases, the charge-density-wave transition temperature is estimated as the temperature at which the phonon energy of the mode driving the structural instability vanishes. The obtained transition temperature in the bulk is around 59 K, in rather good agreement with experiments, and it is just slightly increased in the single-layer limit to 73 K, showing the weak dependence of the transition on dimensionality. Environmental factors could motivate the disagreement between the transition temperatures reported by experiments. Our analysis also demonstrates the predominance of ionic fluctuations over electronic ones in the melting of the charge-density-wave order.

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.

Straintronic effect for superconductivity enhancement in Li-intercalated bilayer MoS2

In this study, ab initio calculations were performed to show that the superconductivity in Li-intercalated bilayer MoS2 could be enhanced by applying either compressive or tensile strain. Moreover, the mechanism for superconductivity enhancement for the tensile strain case was found to be different than that of the compressive strain case. Enhanced electron phonon coupling (EPC) under tensile strain could be explained by an increase in the nesting function involved with the change in the Fermi surface topology in a wide range of Brillouin zones. The superconducting transition temperature T c of 0.46 K at zero strain increased up to 9.12 K under a 6.0% tensile strain. Meanwhile, the enhancement in compressive strain was attributed to the increase in intrinsic electron phonon matrix elements. Furthermore, the contribution from interband scattering was large, which suggested the importance of electron pockets on the Fermi surface. Finally, 80% of the total EPC (λ = 0.98) originated from these pockets and the estimated T c was 13.50 K.

Visualization of Multifractal Superconductivity in a Two-Dimensional Transition Metal Dichalcogenide in the Weak-Disorder Regime

Eigenstate multifractality is a distinctive feature of noninteracting disordered metals close to a metal-insulator transition, whose properties are expected to extend to superconductivity. While multifractality in three dimensions (3D) only develops near the critical point for specific strong-disorder strengths, multifractality in 2D systems is expected to be observable even for weak disorder. Here we provide evidence for multifractal features in the superconducting state of an intrinsic, weakly disordered single-layer NbSe₂ by means of low-temperature scanning tunneling microscopy/spectroscopy. The superconducting gap, characterized by its width, depth, and coherence peaks' amplitude, shows a characteristic spatial modulation coincident with the periodicity of the quasiparticle interference pattern. The strong spatial inhomogeneity of the superconducting gap width, proportional to the local order parameter in the weak-disorder regime, follows a log-normal statistical distribution as well as a power-law decay of the two-point correlation function, in agreement with our theoretical model. Furthermore, the experimental singularity spectrum f(α) shows anomalous scaling behavior typical from 2D weakly disordered 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.

Enhanced Optoelectronic Performance of CVD-Grown Metal-Semiconductor NiTe2/MoS2 Heterostructures

Van der Waals (vdW) heterostructures are the fundamental blocks for two-dimensional (2D) electronic and optoelectronic devices. In this work, a high-quality 2D metal-semiconductor NiTe₂/MoS₂ heterostructure is prepared by a two-step chemical vapor deposition (CVD) growth. The back-gated field-effect transistors (FETs) and photodetectors based on the heterostructure show enhanced electronic and optoelectronic performance than that of a pristine MoS₂ monolayer, owing to the better heterointerface in the former device. Especially, this photodetector based on the metal-semiconductor heterostructure shows 3 orders faster rise time and decay time than that of the pristine MoS₂ under the same fabrication procedure. The enhancement of electronic behavior and optoelectronic response by the epitaxial growth of metallic vdW layered materials can provide a new method to improve the performance of optoelectronic devices.

Thickness-dependent magnetotransport properties in 1T VSe2 single crystals prepared by chemical vapor deposition

Two-dimensional (2D) metallic transition metal dichalcogenides (TMDs) exhibit fascinating quantum effects, such as charge-density-wave (CDW) and weak antilocalization (WAL) effect. Herein, low temperature synthesis of 1T phase VSe₂ single crystals with thickness ranging from 3 to 41 nm by chemical vapor deposition (CVD) is reported. The VSe₂ shows a decreasing phase transition temperature of the CDW when the thickness is decreased. Moreover, low-temperature magnetotransport measurements demonstrate a linear positive and non-saturating magnetoresistance (MR) of 35% from a 35 nm thick VSe₂ at 15 T and 2 K due to CDW induce mobility fluctuations. Surprisingly, Kohler's rule analysis of the MR reveals the non-applicability of Kohler's rule for temperature above 50 K indicating that the MR behavior cannot be described in terms of semiclassical transport on a single Fermi surface with a single scattering time. Furthermore, WAL effect is observed in the 4.2 nm thick VSe₂ at low magnetic fields at 2 K, revealing the contribution of the quantum interference effect at the 2D limit. The phase coherence length [Formula: see text] and spin-orbit scattering length [Formula: see text] were determined to be 73 nm and 18 nm at 2 K, respectively. Our work opens new avenues to study the fundamental quantum phenomena in CVD-deposited TMDs.

Influence of pressure on the transport, magnetic, and structural properties of superconducting Cr0.0009NbSe2 single crystal

We investigate the superconducting critical current density (J _c), transition temperature (T _c), and flux pinning properties under hydrostatic pressure (P) for Cr_0.0009NbSe₂ single crystal. The application of P enhances T _c in both electrical resistivity (∼0.38 K GPa^-1: 0 ≤ P ≤ 2.5 GPa) and magnetization (∼0.98 K GPa^-1: 0 ≤ P ≤ 1 GPa) measurements, which leads to a monotonic increase in J _c and flux pinning properties. The field-dependent J _c at various temperatures under P is analyzed within the collecting pinning theory and it shows that δT _c pinning is the crossover to δl pinning above the critical pressure (P _c ∼0.3 GPa). Our systematic analysis of the flux pinning mechanism indicates that both the density of pinning centers and pinning forces greatly increase with the application of P, which leads to an enhancement in the vortex state. Structural studies using synchrotron X-ray diffraction under pressure illustrate a stable hexagonal phase without any significant impurity phase and lattice parameter reduction with P shows highly anisotropic nature.

Inelastic Electron Tunneling in 2H-Ta_{x}Nb_{1-x}Se_{2} Evidenced by Scanning Tunneling Spectroscopy

We report a detailed study of tunneling spectra measured on 2H-Ta_{x}Nb_{1-x}Se_{2} (x=0∼0.1) single crystals using a low-temperature scanning tunneling microscope. The prominent gaplike feature, which has not been understood for a long time, was found to be accompanied by some "in-gap" fine structures. By investigating the second-derivative spectra and their temperature and magnetic field dependencies, we were able to prove that inelastic electron tunneling is the origin of these features and obtain the Eliashberg function of 2H-Ta_{x}Nb_{1-x}Se_{2} at an atomic scale, providing a potential way to study the local Eliashberg function and the phonon spectra of the related transition-metal dichalcogenides.