Charge Density Wave Ground State Research Articles

Phonon mixing in the charge density wave state of ScV6Sn6

Kagomé metals are widely recognized, versatile platforms for exploring topological properties, unconventional electronic correlations, magnetic frustration, and superconductivity. In the RV6Sn6 family of materials (R = Sc, Y, Lu), ScV6Sn6 hosts an unusual charge density wave ground state as well as structural similarities with the AV3Sb5 system (A = K, Cs, Rb). In this work, we combine Raman scattering spectroscopy with first-principles lattice dynamics calculations to reveal phonon mixing processes in the charge density wave state of ScV6Sn6. In the low temperature phase, we find at least four new peaks in the vicinity of the V-containing totally symmetric mode near 240 cm−1 suggesting that the density wave acts to mix modes of P6/mmm and Rbar{3}m symmetry - a result that we quantify by projecting phonons of the high symmetry state onto those of the lower symmetry structure. We also test the stability of the short-range ordered density wave state under compression and propose that both physical and chemical pressure quench the effect. We discuss these findings in terms of symmetry and the structure-property trends that can be unraveled in this system.

Impact of Sb degrees of freedom on the charge density wave phase diagram of the kagome metal CsV3Sb5

Elucidating the microscopic mechanisms responsible for the charge density wave (CDW) instability of the AV$_3$Sb$_5$ (A=Cs, K, Rb) family of kagome metals is critical for understanding their unique properties, including superconductivity. In these compounds, distinct CDW phases with wave-vectors at the $M$ and $L$ points are energetically favorable, opening the possibility of tuning the type of CDW order by appropriate external parameters. Here, we shed light on the CDW landscape of CsV$_3$Sb$_5$ via a combination of first-principles calculations and phenomenology, which consists of extracting the coefficients of the CDW Landau free-energy expansion from density functional theory. We find that while the main structural distortions of the kagome lattice in the staggered tri-hexagonal CDW phase are along the nearest-neighbor V-V bonds, distortions associated with the Sb ions play a defining role in the energy gain in this and all other CDW states. Moreover, the coupling between ionic displacements from different unit cells is small, thus explaining the existence of multiple CDW instabilities with different modulations along the c-axis. We also investigate how pressure and temperature impact the CDW phase of CsV$_3$Sb$_5$. Increasing pressure does not change the staggered tri-hexagonal CDW ground state, even though the $M$-point CDW instability disappears before the $L$-point one, a behavior that we attribute to the large nonlinear coupling between the order parameters. Upon changing the temperature, we find a narrow regime in which another transition can take place, toward a tri-hexagonal Star-of-David CDW phase. We discuss the implications of our results by comparing them with experiments on this compound.

Adiabatic preparation of fractional Chern insulators from an effective thin-torus limit

We explore the quasi one-dimensional (thin torus, or TT) limit of fractional Chern insulators (FCIs) as a starting point for their adiabatic preparation in quantum simulators. Our approach is based on tuning the hopping amplitude in one direction as an experimentally amenable knob to dynamically change the effective aspect ratio of the system. Similar to the TT limit of fractional quantum Hall (FQH) systems in the continuum, we find that the hopping-induced TT limit adiabatically connects the FCI state to a trivial charge density wave (CDW) ground state. This adiabatic path may be harnessed for state preparation schemes relying on the initialization of a CDW state followed by the adiabatic decrease of a hopping anisotropy. Our findings are based on the calculation of the excitation gap in a number of FCI models, both on a lattice and consisting of coupled wires. By analytical calculation of the gap in the limit of strongly anisotropic hopping, we show that its scaling is compatible with the preparation of large size FCIs for sufficiently large hopping anisotropy. Our numerical simulations in the framework of exact diagonalization explore the full anisotropy range to corroborate these results.

Electronic states of domain walls in commensurate charge density wave ground state and mosaic phase in 1T -TaS2

Domain walls (DWs) in the charge-density-wave (CDW) Mott insulator 1T-TaS2 have unique localized states, which play an important role in exploring the electronic properties of the material. However, the electronic states in DWs in 1T-TaS2 have not been clearly understood, mostly due to the complex structures, phases, and interlayer stacking orders in the DW areas. Here, we explored the electronic states of DWs in the large-area CDW phase and mosaic phase of 1T-TaS2 by scanning tunneling spectroscopy. Due to the different densities of DWs, the electronic states of DWs show distinct features in these phases. In the large area CDW phase, both the domain and the DWs (DW1, DW2, DW4) have zero conductance at the Fermi level; while in the mosaic phase, they can be metallic or insulating depending on their environments. In areas with a high density of DWs, some electronic states were observed both on the DWs and within the domains, indicating delocalized states over the whole region. Our work contributes to further understanding of the interplay between CDW and electron correlations in 1T-TaS2.

Interplay of charge ordering and superconductivity in two-dimensional 2H group V transition-metal dichalcogenides

Layered $2H\text{\ensuremath{-}}T{X}_{2}$ with $T=\mathrm{Nb}$, Ta and $X=\mathrm{S}$, Se exhibit rich dimensionality effects on charge density wave (CDW) order and superconductivity, but comprehensive understanding of these correlated quantum phases in the two-dimensional limit of $2H\text{\ensuremath{-}}T{X}_{2}$ is still lacking, hindering their practical applications. Here, I calculate from first principles the phonon linewidth and bare susceptibility to study the origin of CDW formation in ${\mathrm{NbSe}}_{2}$, ${\mathrm{TaS}}_{2}$, ${\mathrm{TaSe}}_{2}$, and ${\mathrm{NbS}}_{2}$ monolayers, analyze their relative CDW strength, and evaluate electron-phonon superconductivity within the fully anisotropic Migdal-Eliashberg theory. A peaked linewidth of the longitudinal acoustic branch and no Fermi-surface nesting around ${\mathbf{q}}_{\mathrm{CDW}}$ indicate CDW instability driven by the wave-vector-dependent electron-phonon coupling. The $3\ifmmode\times\else\texttimes\fi{}3$ CDW ground state favors a distinct hollow-centered clustering of transition-metal atoms, with larger distortion amplitude in ${\mathrm{NbSe}}_{2}$ and ${\mathrm{TaSe}}_{2}$ than in ${\mathrm{TaS}}_{2}$ and ${\mathrm{NbS}}_{2}$. Superconducting results of the monolayer CDW phase prove that strong spin-orbit coupling effects are dominant in determining the critical temperature ${T}_{\mathrm{c}}$ of each system via modifying both the strength and anisotropic extent of electron-phonon interactions. I also recapitulate measured opposite thickness dependencies of ${T}_{\mathrm{c}}$ in $\mathrm{Nb}{X}_{2}$ and $\mathrm{Ta}{X}_{2}$, and show that whether superconductivity is weakened or enhanced in monolayer limit relies on specific evolution of CDW-modulated Fermi-level density of states and electron-phonon matrix elements under reduced dimensionality. This work lays the foundation for applications of $2H$ group V transition-metal dichalcogenides in versatile nanoscale quantum devices.

Strain engineering and the hidden role of magnetism in monolayer VTe2.

Two-dimensional transition-metal dichalcogenides have attracted great attention recently. Motivated by a recent study of crystalline bulk VTe2, we theoretically investigated the spin-charge-lattice interplay in monolayer VTe2. To understand the controversial experimental reports on several different charge density wave ground states, we paid special attention to the 'hidden' role of antiferromagnetism as its direct experimental detection may be challenging. Our first-principles calculations show that the 4 × 1 charge density wave and the corresponding lattice deformation are accompanied by the 'double-stripe' antiferromagnetic spin order in its ground state. This phase has not only the lowest total energy but also dynamic phonon stability, which supports a group of previous experiments. Interestingly enough, this ground state is stabilized only by assuming the underlying spin order. By noticing this intriguing and previously unknown interplay between magnetism and other degrees of freedom, we further suggest a possible strain engineering. By applying tensile strain, monolayer VTe2 exhibits a phase transition first to a different charge density wave phase and then eventually to a ferromagnetically ordered one.

Topological charge density waves at half-integer filling of a moiré superlattice

At partial filling of a flat band, strong electronic interactions may favor gapped states harboring emergent topology with quantized Hall conductivity. Emergent topological states have been found in partially filled Landau levels and Hofstadter bands; in both cases, a large magnetic field is required to engineer the underlying flat band. The recent observation of quantum anomalous Hall effects (QAH) in narrow band moir\'e systems has led to the theoretical prediction that such phases may be realized even at zero magnetic field. Here we report the experimental observation of insulators with Chern number $C=1$ in the zero magnetic field limit at $\nu=3/2$ and $7/2$ filling of the moir\'e superlattice unit cell in twisted monolayer-bilayer graphene (tMBG). Our observation of Chern insulators at half-integer values of $\nu$ suggests spontaneous doubling of the superlattice unit cell, in addition to spin- and valley-ferromagnetism. This is confirmed by Hartree-Fock calculations, which find a topological charge density wave ground state at half filling of the underlying $C=2$ band, in which the Berry curvature is evenly partitioned between occupied and unoccupied states. We find the translation symmetry breaking order parameter is evenly distributed across the entire folded superlattice Brillouin zone, suggesting that the system is in the flat band, strongly correlated limit. Our findings show that the interplay of quantum geometry and Coulomb interactions in moir\'e bands allows for topological phases at fractional superlattice filling that spontaneously break time-reversal symmetry, a prerequisite in pursuit of zero magnetic field phases harboring fractional statistics as elementary excitations or bound to lattice dislocations.

X-ray Studies of the CDW Ground State and Excitations in High-TC Cuprates

Charge density wave (CDW) in high-TC cuprates has drawn a lot of attention in the field. In this article, we review some of our recent x-ray studies about the ground state and the excitation spectr...

Ultrafast excitation and topological soliton formation in incommensurate charge density wave states

Topological soliton is a nonperturbative excitation in commensurate density wave states and connects degenerate ground states. In incommensurate density wave states, ground states are continuously degenerate and topological soliton is reckoned to be smoothly connected to the perturbative phason excitation. We study the ultrafast nonequilibrium dynamics due to photoexcited electron-hole pair in a one-dimensional chain with an incommensurate charge density wave ground state. Time-resolved evolution reveals both perturbative excitation of collective modes and nonperturbative topological phase transition due to creating novel topological solitons, where the continuous complex order parameter with amplitude and phase is essential. We identify the nontrivial phase-winding solitons in the complex plane unique to this nonequilibrium state and capture it by a low-energy effective model. The perturbative temporal gap oscillation and the solitonic in-gap states enter the optical conductivity absorption edge and the spectral density related to spectroscopic measurement, providing concrete connections to real experiments.

Charge Density Waves and Electronic Properties of Superconducting Kagome Metals.

Kagome metals AV_{3}Sb_{5} (A=K, Rb, and Cs) exhibit intriguing superconductivity below 0.9∼2.5 K, a charge density wave (CDW) transition around 80∼100 K, and Z_{2} topological surface states. The nature of the CDW phase and its relation to superconductivity remains elusive. In this work, we investigate the electronic and structural properties of CDW by first-principles calculations. We reveal an inverse Star of David deformation as the 2×2×2 CDW ground state of the kagome lattice. The kagome lattice shows softening breathing-phonon modes, indicating the structural instability. However, electrons play an essential role in the CDW transition via Fermi surface nesting and van Hove singularity. The inverse Star of David structure agrees with recent experiments by scanning tunneling microscopy (STM). The CDW phase inherits the nontrivial Z_{2}-type topological band structure. Further, we find that the electron-phonon coupling is too weak to account for the superconductivity T_{c} in all three materials. It implies the existence of unconventional pairing of these kagome metals. Our results provide essential knowledge toward understanding the superconductivity and topology in kagome metals.

Charge Density Waves in Electron-Doped Molybdenum Disulfide

We present the discovery of a charge density wave (CDW) ground state in heavily electron-doped molybdenum disulfide (MoS2). This is the first observation of a CDW in any d2 (column 6) transition metal dichalcogenide (TMD). The band structure of MoS2 is distinct from the d0 and d1 TMDs in which CDWs have been previously observed, facilitating new insight into CDW formation. We demonstrate a metal–insulator transition at 85 K, a 25 meV gap at the Fermi level, and two distinct CDW modulations, (2√3 × 2√3) R30° and 2 × 2, attributable to Fermi surface nesting (FSN) and electron–phonon coupling (EPC), respectively. This simultaneous exhibition of FSN and EPC CDW modulations is unique among observations of CDW ground states, and we discuss this in the context of band folding. Our observations provide a route toward the resolution of controversies surrounding the origin of CDW modulations in TMDs.

Non-equilibrium charge density wave ground state of quasi-two-dimensional rare-earth tritelluride TbTe3

We have studied the time dependence of the relaxation of the non-equilibrium charge density wave (CDW) toward an equilibrium ground state in TbTe3 when the sample is cooled through the Peierls transition temperature under an electric field down to a given temperature, Texp. We show that when cooled at zero electric field or a value less than the threshold one, Et, for depinning the CDW at Texp, the CDW equilibrium ground state has a single phase in the sample volume. When cooled with an electric field higher than Et, the CDW ground state is in a frozen glass state, which can be destabilized only by reducing the electric field below Et. We tentatively interpret these results by the interaction of the CDW with a well ordered Te–Te discommensuration network.

Momentum-dependent electron-phonon coupling in charge density wave systems

Many charge density wave (CDW) systems exhibit $q(T)$ electron-hole modulations continuously varying with $T$ and saturating upon cooling at an incommensurate value even if the maximum occurring in the electron-hole Lindhard response does not exhibit such a thermal shift. Using a simple RPA argument we show that the experimental $q(T)$ can be understood if the electron-phonon coupling (EPC) $g(q)$, necessary to set coupled electronic and structural modulations, is momentum dependent. In this analysis, the sense of variation of $q(T)$ depends upon the sign of $\frac{\ensuremath{\partial}g(q)}{\ensuremath{\partial}q}$ and its amplitude of thermal variation is controlled by the electron-hole coherence length (or CDW rigidity) in the modulation direction. This model quantitatively accounts for the thermal dependence of $q(T)$ in the one-dimensional (1D) CDW system ${\mathrm{K}}_{0.3}\mathrm{Mo}{\mathrm{O}}_{3}$ (blue bronze) both in its CDW ground state and in its pretransitional CDW fluctuation regime. We suggest that such a general analysis can be extended to account for the $q(T)$ dependence observed in other 1D and 2D CDW systems such as the transition metal di- and trichalcogenides as well as the lanthanide and rare-earth tritellurides. Using a detailed analysis of the low frequency phonon spectrum of the blue bronze, we then propose a new scenario for the $q$ dependent EPC, where $g(q)$ is due to a momentum-dependent hybridization between the critical phonon branch bearing the Kohn anomaly and other low-lying phonon branches. This allows obtaining a sign of $\frac{\ensuremath{\partial}g(q)}{\ensuremath{\partial}q}$ in agreement with that deduced from the analysis of $q(T)$. Finally, we propose that similar hybridization effects could also be relevant for other 1D and 2D CDW systems exhibiting a thermally dependent modulation.

Superconducting phase diagrams of S-doped 2H−TaSe2 under hydrostatic pressure

We report the hydrostatic pressure effect on the charge density wave (CDW) and superconductivity (SC) of single-crystal $2H\text{\ensuremath{-}}\mathrm{Ta}{\mathrm{Se}}_{2\text{\ensuremath{-}}x}{\mathrm{S}}_{x}$ by measuring the electrical resistivity and ac magnetic susceptibility under various pressures up to 12.5 GPa. Different from the previously reported phase diagrams of CDW and SC as a function of the Se/S substitution and/or the uniaxial pressure, the superconducting transition temperature (${T}_{\mathrm{c}}$) was found to increase monotonously with increasing the pressure at first and then reaches a maximum constant above a critical pressure ${P}_{\mathrm{c}}$. The maximal ${{T}_{\mathrm{c}}}^{\mathrm{max}}$ is nearly two times higher than the optimal ones via the S doping and decreases with further increasing the S-doping level. High-pressure ac magnetic susceptibility demonstrates that the superconducting volume quickly increases along with the gradual suppression of CDW and reaches nearly 1 above ${P}_{\mathrm{c}}$, which provides clear evidences for pressure-induced crossover from CDW to bulk SC. High-pressure x-ray diffraction proves that there is no structural phase transition in $2H\text{\ensuremath{-}}\mathrm{Ta}{\mathrm{Se}}_{2}$ up to 20 GPa; although the dependence of ${T}_{\mathrm{c}}$ on the volume shows distinct behaviors at the initial S doping and the applied physical pressure, they both collapse into a universal curve upon the further volume shrinkage. This indicates that both physical pressure and the S-doping-induced chemical pressure can melt CDW and enhance SC through lattice contraction consistently. In comparison with the effects of hydrostatic pressure, the S doping can destruct the CDW ground state faster due to the presence of chemical disorders, which also restrict the further enhancement of ${T}_{\mathrm{c}}$ after the collapse of CDW.

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.

Two-dimensional chiral stacking orders in quasi-one-dimensional charge density waves

Chirality manifests in various forms in nature. However, there is no evidence of the chirality in one-dimensional charge density wave (CDW) systems. Here, we have explored the chirality among quasi-one-dimensional CDW ground states with the aid of scanning tunneling microscopy, symmetry analysis, and density functional theory calculations. We discovered three distinct chiralities emerging in the form of two-dimensional chiral stacking orders composed of degenerate CDW ground states: right-, left-, and nonchiral stacking orders. Such chiral stacking orders correspond to newly introduced chiral winding numbers. Furthermore, we observed that these chiral stacking orders are intertwined with chiral vortices and chiral domain walls, which play a crucial role in engineering the chiral stacking orders. Our findings suggest that the unexpected chiral stacking orders can open a way to investigate the chirality in CDW systems, which can lead to diverse phenomena such as circular dichroism depending on chirality.

Charge density waves beyond the Pauli paramagnetic limit in 2D systems

Two-dimensional materials are ideal candidates to host Charge Density Waves (CDWs) that exhibit paramagnetic limiting behavior, similar to the well-known case of superconductors. Here, we study how CDWs in two-dimensional systems can survive beyond the Pauli limit when they are subjected to a strong magnetic field by developing a generalized mean-field theory of CDWs under Zeeman fields that includes incommensurability, imperfect nesting, and temperature effects and the possibility of a competing or coexisting Spin Density Wave (SDW) order. Our numerical calculations yield rich phase diagrams with distinct high-field phases above the Pauli limiting field. For perfectly nested commensurate CDWs, a q-modulated CDW phase that is completely analogous to the superconducting Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase appears at high fields. In the more common case of imperfect nesting, the commensurate CDW ground state undergoes a series of magnetic-field-induced phase transitions first into a phase where commensurate CDW and SDW coexist and subsequently into another phase where CDW and SDW acquire a q-modulation that is, however, distinct from the pure FFLO CDW phase. The commensurate CDW + SDW phase occurs for fields comparable to but less than the Pauli limit and survives above it. Thus, this phase provides a plausible mechanism for the CDW to survive at high fields without the need for forming the more fragile FFLO phase. We suggest that the recently discovered 2D materials like the transition metal dichalcogenides offer a promising platform for observing such exotic field-induced CDW phenomena.

Defect-Selective Charge-Density-Wave Condensation in 2H-NbSe_{2}.

Defects have been known to substantially affect quantum states of materials including charge density wave (CDW). However, the microscopic mechanism of the influence of defects is often elusive due partly to the lack of atomic scale characterization of defects themselves. We investigate native defects of a prototypical CDW material 2H-NbSe_{2} and their microscopic interaction with CDW. Three prevailing types of atomic scale defects are classified by scanning tunneling microscope, and their atomic structures are identified by density functional theory calculations as Se vacancies and Nb intercalants. Above the transition temperature, two distinct CDW structures are found to be induced selectively by different types of defects. This intriguing phenomenon is explained by competing CDW ground states and local lattice strain fields induced by defects, providing a clear microscopic mechanism of the defect-CDW interaction.

Toward the equilibrium ground state of the charge density waves in rare-earth tritellurides

We show that the charge density wave (CDW) ground state below the Peierls transition temperature, $T_{CDW}$, of rare-earth tritellurides is not at its equilibrium value, but depends on the time where the system was kept at a fixed temperature below $T_{CDW}$. This ergodicity breaking is revealed by the increase of the threshold electric field for CDW sliding which depends exponentially on time. We tentatively explain this behavior by the reorganization of the oligomeric (Te$_x$)$^{2-}$ sequence forming the CDW modulation.

Nonadiabatic quantum fluctuations in the neutral ground state of tetrathiafulvalene- p -chloranil

A nonadiabatic resonating Hartree-Fock method is applied to the one-dimensional extended Hubbard model with a staggered site-diagonal potential and Su-Schrieffer-Heeger-type electron-phonon coupling to clarify the lattice and electronic structures in the charge-density-wave (CDW) ground state. It is shown that small spin-Peierls-type domains and large dimerized CDW domains appear in the main CDW state having an equidistant lattice. Interference of breathing and translational motions of these domains constitutes quantum fluctuations. The natures of different domains are analyzed by L\owdin-Feshbach partitioning method. The present results, indicating the domains in the CDW ground state as quantum fluctuations, are consistent with the experimentally observed THz electric-field-induced changes in the electronic dipole moment in the neutral phase of tetrathiafulvalene-$p$-chloranil (TTF-CA).