Prediction of Potential Charge Density Wave in Monolayer H‐NbS 2 by First‐Principles Calculations

This work presents a detailed study of the electronic structure, phonon dispersion, Z2 invariant calculation, and Fermi surface of the newly discovered kagome superconductor CsV3Sb5, using density functional theory. The phonon dispersion in the pristine state reveals two negative modes at the M and L points of the Brillouin zone, indicating lattice instability. CsV3Sb5 transitions into a structurally stable 2 × 2 × 1 charge density wave (CDW) phase, confirmed by positive phonon modes. The electronic band structure shows several Dirac points near the Fermi level, with a narrow gap opening due to spin–orbit coupling (SOC), although the effect of SOC on other bands is minimal. In the pristine phase, this material exhibits a quasi-2D cylindrical Fermi surface, which undergoes reconstruction in the CDW phase. We calculated quantum oscillation frequencies using Onsager’s relation, finding good agreement with experimental results in the CDW phase. To explore the topological properties of CsV3Sb5, we computed the Z2 invariant in both pristine and CDW phases, resulting in a value of (ν0; ν1ν2ν3) = (1; 000), suggesting the strong topological nature of this material. Our detailed analysis of phonon dispersion, electronic bands, Fermi surface mapping, and Z2 invariant provides insights into the topological properties, CDW order, and unconventional superconductivity in AV3Sb5 (A = K, Rb, and Cs).

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.

Monolayer transition metal dichalcogenide VTe2 exhibits multiple charge density wave (CDW) phases, mainly (4 × 4) and (4 × 1). Here we report facile dynamic and tens-of-nanometer scale switching between these CDW phases with gentle bias pulses in scanning tunneling microscopy. Bias pulses purposely stimulate a reversible random CDW symmetry change between the isotropic (4 × 4) and anisotropic (4 × 1) CDWs, as well as CDW phase slips and rotation. The switching threshold of ∼1.0 V is independent of bias polarity, and the switching rate varies linearly with the tunneling current. Density functional theory calculations indicate that a coherent CDW phase switching incurs an energy barrier of ∼2.0-3.0 eV per (4 × 4) unit cell. While there is a challenge in understanding the observed large-area CDW random fluttering, we provide some possible explanations. The ability to manipulate electronic CDW phases sheds new light on tailoring CDW properties on demand.

Lifshitz transition was proposed to explain a change of the topology structure in a Fermi surface induced by continuous lattice deformation without symmetry breaking since 1960. It is well known that the anomalies of the kinetic coefficients (the coefficient of heat conduction and electrical conductivity, viscosity, sound absorption, etc.) are usually closely connected with the Lifshitz transition behavior. 2H-TaS2 is a typical representative to study its anomalies of temperature dependence of heat capacity, resistivity, Hall effect, and magnetic susceptibility. Its geometrical structure of the charge density wave (CDW) phase and layer number dependence of carrier-sign alternation upon cooling in the Hall measurements have not been well understood. The geometrical structure (T-Ts) of the CDW phase was predicted through first principles calculations for bulk and mono-layer 2H-TaS2. Driven by electron-lattice coupling, Ta atoms contract to form a partially gapped CDW phase. The CDW phase has a larger average interlayer separation of S-S atoms in the adjacent two layers compared with the metal phase, which results in a weaker chemical bonding among S-S atoms in the adjacent two layers and then a narrower bandwidth of the energy band. The narrower bandwidth of the energy band leads to a larger density of states (DOS) in the out-of-plane direction above the Fermi level for the CDW phase. As the Fermi level continually drops from the DOS region with a negative slope to that with a positive slope on cooling, the reversal of the p → n type carrier and the pocket-vanishing-type Lifshitz transition occur in the bulk 2H-TaS2. However, the Fermi level slightly drops by 6 meV and happens to be at the positions of pseudo band gaps, so the reduction of in-plane DOS and total DOS is responsible for the always p-type carrier in the mono-layer samples. Our CDW vector of the k-space separation between two saddle points is QSP ≈ 0.62 GK and can provide a theoretical support for the "saddle-point" CDW mechanism proposed by Rice and Scott. Our theoretical explanation gives a new understanding of both Lifshitz transition for symmetry breaking and reversal for the p-n carrier sign in the Hall measurements in various two-dimensional transition metal disulfides.

Dimensionality control provides a route to adjust physical properties in van der Waals layered materials. Bulk CuTe forming a layered structure exhibits a charge density wave (CDW) phase in Te chains with a periodicity of 5×1×2 at temperature below 335 K. The stability of the CDW state in a CuTe monolayer, however, has been investigated neither experimentally nor theoretically. Here, we report the theoretical prediction for the CDW phase in a CuTe monolayer using the first principles calculations. Similarly to its bulk structure, we find the phonon soft mode at q = (0.4, 0.0), indicating structural instability, which only appears with the correlation effect on Cu. The role of Coulomb correlations in driving the CDW transition suggests an electron-electron correlation and an electron-phonon interaction as the origin of the CDW instability. We expect the periodicity of the CDW phase in the CuTe monolayer to be 5×1. We show, by reducing the interlayer interactions, that tuning of the CDW modulation may be possible, as demonstrated by the modulation pattern in quasi-one-dimensional Te chains being different from that in the bulk counterpart.

Exploring topological phases in interacting systems is a challenging task. We investigate many-body topological physics of interacting fermions in an extended Su-Schrieffer-Heeger (SSH) model, which extends the two sublattices of SSH model into four sublattices and thus is dubbed SSH4 model, based on the density-matrix renormalization-group numerical method. The interaction-driven phase transition from topological insulator to charge density wave (CDW) phase can be identified by analyzing the variations of entanglement spectrum, entanglement entropies, energy gaps, CDW order parameter, and fidelity. We map the global phase diagram of the many-body ground state, which contains nontrivial topological insulator, trivial insulator and CDW phases, respectively. In contrast to interacting SSH model, in which the phase transitions to the CDW phase are argued to be first-order phase transitions, the phase transitions between the CDW phase and topologically trivial/nontrivial phases are shown to be continuous phase transitions. Finally, we {also} show the phase diagram of interacting spinful SSH4 model, where the attractive (repulsive) on-site spin interaction amplifies (suppresses) the CDW phase. The models analyzed here can be implemented with ultracold atoms on optical superlattices.

The kagome superconductor CsV3Sb5 hosts a variety of charge density wave (CDW) phases, which play a fundamental role in the formation of other exotic electronic instabilities. However, identifying the precise structure of these CDW phases and their intricate relationships remain the subject of intense debate, due to the lack of static probes that can distinguish the CDW phases with identical spatial periodicity. Here, we unveil the out-of-equilibrium competition between two coexisting 2 × 2 × 2 CDWs in CsV3Sb5 harnessing time-resolved X-ray diffraction. By analyzing the light-induced changes in the intensity of CDW superlattice peaks, we demonstrate the presence of both phases, each displaying a significantly different amount of melting upon excitation. The anomalous light-induced sharpening of peak width further shows that the phase that is more resistant to photo-excitation exhibits an increase in domain size at the expense of the other, thereby showcasing a hallmark of phase competition. Our results not only shed light on the interplay between the multiple CDW phases in CsV3Sb5, but also establish a non-equilibrium framework for comprehending complex phase relationships that are challenging to disentangle using static techniques.

Vanadium ditelluride (VTe2) has been intensively explored to understand the charge density wave (CDW) phase and its connection to magnetic properties. Here, we conduct a systematic study to understand the fine structure of CDW via scanning tunneling microscopy (STM) combined with density functional theory (DFT) calculations. STM topograph at 79 K shows that a CDW phase in VTe2 has a stripe modulation with 3 × 1 periodicity, following the double zigzag chain of distorted Te lattices. Interestingly, the 3 × 1 CDW modulation undergoes contrast inversion between filled and empty state topographs. Atomistic features and contrast changes of CDW observed in STM are clearly reproduced in our DFT simulation images. Charge distribution calculation indicates that the spatial extension and density of Te 5p orbitals have strong variations with filled and empty states, explaining the fine structure of 3 × 1 CDW in VTe2. Our finding provides an inspiring insight to further research on the less explored electronic structure of VTe2.

Transition metal dichalcogenides are promising candidates to show long-range ferromagnetic order in the single-layer limit. Based on ab initio calculations, we report the emergence of a charge density wave (CDW) phase in monolayer 1T-CrTe$_2$. We demonstrate that this phase is the ground state in the single-layer limit at any strain value. We obtain an optical phonon mode of $1.96$ THz that connects CDW phase with the undistorted 1T phase. Localization of the $a_{1g}$ orbital of CrTe$_2$ produces an out-of-plane orientation of the magnetic moments, circumventing the restrictions of the Mermin-Wagner theorem and producing ferromagnetic long-range order in the two-dimensional limit. This orbital-localization is enhanced by the CDW phase. Tensile strain also increases the localization of this orbital driving the system to become ordered. CrTe$_2$ becomes an example of a material where the CDW phase produces the stabilization of the long-range ferromagnetic order. Our results show that both strain and phase switching are mechanisms to control the 2D ferromagnetic order of CrTe$_2$.

Interplay between the charge density wave phase and a pseudogap under antiferromagnetic correlations

The role of dynamics in the charge density wave (CDW) phases of 1T-TaS2, especially in accessing metastable phases, is still under scrutiny. We investigated cooling rate dependence on low-temperature CDW phases in this material by scanning tunneling microscopy and x-ray diffraction. In the majority of cases, we found the typical low temperature commensurate CDW and identified no other differences between samples that were fast cooled vs slow cooled from room temperature. In rare cases (1/18 STM experiments and 1/30 XRD experiments), we found a multi-domain structure in the low temperature CDW. The domain structure and metallic behavior revealed by STM and STS measurements of this phase are in excellent agreement with the previously reported metallic mosaic CDW phase.

The charge density wave (CDW) phase in bulk and exfoliated crystals of transition metal dichalcogenides was studied by scanning tunneling microscopy (STM). In exfoliated flakes of VSe2 we found a striking non-monotonic evolution of the CDW transition temperature as function of thickness derived from the analysis of the real space charge modulation amplitude imaged by STM. This finding lifts the contradiction of previous measurements. In bulk crystals of VSe2, CuxTiSe2 and NbSe2 we examined the CDW phase by our original method to extract the local amplitude, wavelength and phase of the CDW. This high resolution spatial mapping of the full complex order parameter provided unprecedented insight into fundamental aspects of CDWs. It experimentally proved that the CDW in these materials consist of three individual charge modulations. Furthermore, it explained the different contrasts often observed in STM images and revealed a rich variety of features like phase domains and topological defects.

In this thesis I investigate the relationship between the charge density wave (CDW) phase and superconductivity in the T-x phase diagram of Cu_xTiSe₂. I find that the incommensurate (IC)-CDW is related to the superconducting phase due to the fact that the former effectively isolates the CDW subsystem degrees of freedom. This increases the symmetry of the electronic populations within the IC-CDW band structure and leave them susceptible to internal instabilities, which in turn give rise to the superconducting phase. Because the correlated properties of these solid-state phases of matter are highly dependent on the crystalline quality of our samples, I also detail the growth of pristine single crystals and utilize several characterization techniques to aid in this purpose. In this portion of the thesis the single crystals are deliberately injected with heat and monitored to deduce the formation of defects through selenium migration. I also confirm the existence of chiral symmetry breaking in the bulk commensurate (C)-CDW phase in TiSe₂ brought about by the cooperation of phonon and exciton degrees of freedom, and also observe chiral character in fluctuations above T_[CDW]. These thermal fluctuations were observed up to 80 K above T_[CDW] via optical signatures of the folded Se-4p band and Raman signatures of the soft L_1^- phonon mode. The suppression of the excitonic degree of freedom with Cu intercalation brings about a quantum phase transition into the IC-CDW at x=0.04. Large quantum fluctuations of the folded Se-4p electronic band were observed at the quantum phase transition where measurements of the phonon system show the onset of incommensuration in the CDW super-lattice. Optical measurements demonstrate a large decoupling of the electron-phonon degrees of freedom within the electronic band structure of the IC-CDW subsystem.

Transition metal dichalcogenides are lamellar materials which can exhibit unique and remarkable electronic behavior due to effects of electron-electron and electron-phonon coupling. Among these materials, 1T-tantalum disulfide (1T-TaS2) has spurred considerable interest, due to its multiple first order phase transitions between different charge density wave (CDW) states. In general, the basic effects of charge density wave formation in 1T-TaS2 can be attributed to in plane re-orientation of Ta-atoms during the phase transitions. Only in recent years, an increasing number of studies has also emphasized the role of interlayer interaction and stacking order as a crucial aspect to understand the specific electronic behavior of 1T-TaS2, especially for technological systems with a finite number of layers. Obviously, continuously monitoring the out of plane expansion of the sample can provide direct inside into the rearrangement of the layer structure during the phase transition. In this letter, we therefore investigate the c-axis lattice discontinuities of 1T-TaS2 by atomic force microscopy (AFM) method under ultra-high vacuum conditions. We find that the c-axis lattice experiences a sudden contraction across the nearly-commensurate CDW (NC-CDW) phase to commensurate CDW (C-CDW) phase transition during cooling, while an expansion is found during the transition from the C-CDW phase to a triclinic CDW phase during heating. Thereby our measurements reveal, how higher order C-CDW phase can favor a more dense stacking. Additionally, our measurements also show subtler effects like e.g. two expansion peaks at the start of the transitions, which can provide further insight into the mechanisms at the onset of CDW phase transitions.

In the vicinity of a competing electronic order, superconductivity emerges within a superconducting dome in the phase diagram, which has been demonstrated in unconventional superconductors and transition-metal dichalcogenides (TMDs), suggesting a scenario where fluctuations or a partial melting of a parent order are essential for inducing superconductivity. Here, we present a contrary example, the two-dimensional (2D) superconductivity in transition-metal carbide can be readily turned into charge density wave (CDW) phases via dilute magnetic doping. Low temperature scanning tunneling microscopy/spectroscopy (STM/STS), transport measurements, and density functional theory (DFT) calculations were employed to investigate Cr-doped superconducting Mo2C crystals in the 2D limit. With ultralow Cr doping (2.7 atom %), the superconductivity of Mo2C is heavily suppressed. Strikingly, an incommensurate density wave (IDW) and a related partially opened gap are observed at a temperature above the superconducting regime. The wave vector of IDW agrees well with the calculated Fermi surface nesting vectors. By further increasing the Cr doping level to 9.4 atom %, a stronger IDW with a smaller periodicity and a larger partial gap appear concurrently. The resistance anomaly implies the onset of the CDW phase. Spatial-resolved and temperature-dependent spectroscopy reveals that such CDW phases exist only in a nonsuperconducting regime and could form long-range orders uniformly. The results provide the understanding for the interplay between charge ordered states and superconductivity in 2D transition-metal carbide.