Patterns and driving forces of dimensionality-dependent charge density waves in 2H-type transition metal dichalcogenides

Charge density wave (CDW) is a startling quantum phenomenon, distorting a metallic lattice into an insulating state with a periodically modulated charge distribution. Astonishingly, such modulations appear in various patterns even within the same family of materials. Moreover, this phenomenon features a puzzling diversity in its dimensional evolution. Here, we propose a general framework, unifying distinct trends of CDW ordering in an isoelectronic group of materials, 2H-MX2 (M = Nb, Ta and X = S, Se). We show that while NbSe2 exhibits a strongly enhanced CDW order in two dimensions, TaSe2 and TaS2 behave oppositely, with CDW being absent in NbS2 entirely. Such a disparity is demonstrated to arise from a competition of ionic charge transfer, electron-phonon coupling, and electron correlation. Despite its simplicity, our approach can, in principle, explain dimensional dependence of CDW in any material, thereby shedding new light on this intriguing quantum phenomenon and its underlying mechanisms.

This work experimentally investigates the CDW transition in thin exfoliated sheets of NbSe2, TaSe2 and TaS2, based upon the emergence of the corresponding amplitude modes in the Raman spectrum. A major finding is that T(CDW) increasing (decreases) with decreasing sheet thickness for NbSe2 (TaSe2 and TaS2).
As there are indeed partially contradictory theoretical predictions and also experimental observations reported in the literature on CDWs in this (and similar) type(s) of material, the present study makes a novel and valuable contribution to this field. The detailed Raman experiments seem to be carried out carefully (it is especially important that the bulk crystals were exfoliated in a glove box and subsequently encapsulated by h-BN), and also the data evaluation looks reasonable.
Furthermore, the three parameter-based model ( fig. 1a) proposed to account for the presence or absence (like in NbS2) of CDW instability, and the thickness dependence of T(CDW) for the three different compounds, makes sense and is -despite its simplicity -in quite good agreement with the experimental observations. Although no fundamentally new theoretical result is provided, the presented model can nevertheless help to further clarify relevant issues on CDWs in 2D materials. Finally, also the mentioned experimental factors (in particular the role of the substrate) are indeed likely to have an influence on T(CDW) in these compounds.
Note: In accordance with Nature Communications format and style, we have rephrased the Abstract and Introduction of our manuscript. We have also added section and subsection titles to meet the editorial requirements. These changes do not affect our original discussions. We appreciate both reviewers for taking the time to evaluate our work, and hope with revisions, described below, we have been able to address all their valuable comments and suggestions.
Dear editor, I have read the manuscript "Patterns and driving forces of dimensionality-dependent charge density waves in 2H-type transition metal dichalcogenides" by Dongjing Lin et al. In the paper, they investigate the phonon modes of TaS 2 , TaSe 2 , NbS 2 and NbSe 2 using Raman spectroscopy. This characterization is carried out both at low and at room temperature and for monolayer, bilayer and bulk materials. These experimental results are supplemented by ab-initio and theoretical analysis. The main goal of the paper is to gain more understanding of the differences in the CDWs observed in these materials. Throughout the paper, NbS 2 is the odd one out: it does not have a CDW at any temperature.
I found the manuscript a pleasure to read. The experimental results are laid out clearly and comprehensively, covering a relevant and diverse set of experimental conditions. Transition metal-dichalcogenides are an active topic of current experimental and theoretical research and this work is a very worthy contribution to the field, both in terms of I believe it should be published in Nature Communications. I do have some questions and remarks, which follow below.
We thank the reviewer for careful reading of our manuscript and constructive comments, enabling us to improve our work. We have taken the reviewer's comments seriously, and have revised our manuscript, accordingly. Specific points are addressed below. We hope with these changes, the reviewer will recommend our manuscript for publication. * Around line 70: This paragraph contains a discussion of mechanisms contributing for or against CDW. However, the reason why this class of materials wants to form a CDW in the first place is left rather open here.
We thank the reviewer for raising this critical issue. We believe that the charge density wave (CDW) in transition metal dichalcogenides (TMDs) stems from two facts. The first is the quasi-two-dimensionality of electronic properties in these systems, which is itself due to the layered nature of their crystal structures. The weak van der Waals bonding between the constituent layers allows each unit to keep its characteristics to a large extent and at the same time let the phonon vibrations to effectively rattle such weakly connected sheets. Secondly, as we have discussed in our paper, metallic TMDs, including 2H-MX 2 compounds studied here, suffer from a deficient charge transfer from the transition metal cations to the chalcogen anions. To compensate for this deficiency, the chalcogen p orbitals try to form additional covalent bonding with each other that when combined with existing phonon vibrations, can propagate a charge density modulation throughout the whole system. This combination, in our opinion, is the key factor in the emergence of CDW in metallic TMDs.
To clarify this, we have added the following explanation in the introduction part of our revised manuscript on page 3: Two constraints facilitate the emergence of CDW instability and its diversity in this group of materials. The first is the fact that TMDs, in general, possess a quasi-2D electronic structure regardless of their thickness. This is due to the weak van der Waals stacking of MX 2 layers, allowing resonance of phonon vibrations along certain crystalline directions. Second, the metallic TMDs, as will be discussed later, exhibit a strong tendency to form additional covalent bonding between their chalcogens to compensate for the deficient charge transfer from the existing transition metals. When combined, these two can modulate the otherwise uniform charge distribution of metallic bands, such that the distorted state is chemically more robust.
* Around line 205: Is this analysis based on the equation λ = 2g 2 /ω phonon ? In that case, it might be useful to include this formula. * More generally, including a clearer definition of the electron-phonon coupling λ in the main text could be helpful, also to put Fig 4b into context.
That is correct. In principle, this is the equation used for treating electron-phonon coupling λ. More precisely, we have used the implementation developed by Poncé et al.
(Reference 50 in the revised manuscript) for this purpose. To clarify this, We have added a detailed description in the "Methods" part of our revised manuscript to elaborate on the methodology used for the calculation of λ. It reads: In this scheme 50 , λ is obtained by summing up the electron-phonon coupling arising from each individual phonon mode ν at all available phonon wave vectors q, i.e. λ = qν λ qν . λ qν is expressed as follows, where ω qν is the phonon frequency for mode ν at the wave vector q; ε nk is the eigenvalue of the n-th Kohn-sham eigenfunction ψ nk at wave vector k in the BZ; ε F is the Fermi level; N (ε F ) corresponds to the density of states per spin at N (ε F ) and Ω BZ is the volume of the BZ. Above, g ν mn (k, q) is the first-order electron-phonon matrix element defined as, with ∂ qν V being the first derivative of the self-consistent potential associated with phonon ω qν .
* The authors have performed ab-initio calculations, for the band structure and the electron-phonon coupling. I would be interested to know about the ab-initio phonon spectra and their relation with the experimentally observed Raman spectra.
Following the reviewer's suggestion, we have included the phonon dispersions for all four monolayer compounds in a new section (Section 2) in our Supplementary information. To connect them with our experimental data, we have added legends in each panel to indicate the corresponding A 1g and E 2g branches, appearing in our Raman spectroscopy measurements as active phonon modes. As can be seen the agreement with experiment is satisfactorily good, and many key features such as reversal of phonon modes in NbSe 2 is well reproduced.
The changes in the main text is as follows: In the first paragraph on page 5 after "As shown in Figure 2c, both the A 1g and E 2g modes are found to have much higher frequencies in S-based 2H-MX 2 compounds than in Se-based ones." we have added Similar results are found for the monolayer samples, also consistent with our first-principles calculations (Supplementary Section 2).
Also, below is the new Supplementary Section 2: 2. Comparison of phonon dispersions Figure S3 summarizes the calculated phonon dispersions for all four monolayer 1H-MX 2 (M=Nb, Ta and X=S, Se) compounds. In a glance, one can notice a systematic difference between the S-based and Se-based compounds. In the former, the optical modes appear at much higher frequencies. Also, the gap between the A 1g and E 2g modes is much larger in the S-based compounds than in the Se-based ones. As described in the main text, such a hardening of phonon modes is an indication that the chemical bonding in S-based compounds is more ionic (and thus stronger) than that in Se-based compounds. Overall, the calculated values for A 1g and E 2g modes at Γ point (i.e. q = (0, 0, 0)) agree reasonably well with our experimental data, as shown in Figure S4. The largest deviation is found for NbS 2 where the calculations underestimate the E 2g mode by ∼ 19%. We attribute this to the localized nature of wave-functions in this material, requiring exchange-correlation approximations beyond standard density functional methods such as Perdew-Burke-Ernzerhof functional used in this study as well as more accurate pseudo-potentials. Nevertheless, our calculations provide a strong confirmation for many key observations in this study, including the fact that NbSe 2 is the only compound in this family in which A 1g mode lies below the E 2g mode (see for example  , 10210-10220) discuss the role of the enviroment in TaS 2 monolayers in more detail. They find that it is not just charge transfer but also hybridization between monolayer and substrate that is responsible for the absence of CDWs on gold substrates.
We thank the reviewer for pointing out this important reference. We have included it and two more related works on 2H-TaS 2 , and expanded the corresponding discussions in the revised version. It now reads: Interestingly, CDWs were found to be absent in epitaxial monolayer TaS 2 on Au(111) substrate 37 , but persist in MBE-grown monolayer TaS 2 on graphene/Ir(111) 38 as well as in the exfoliated monolayer studied here. These findings provide further evidence that the CDW formation in such atomically thin TMDs is, in general, highly susceptible to the surrounding environment, which could be a neighboring layer or even a substrate. As we discussed earlier, the deficient charge transfer in metallic TMDs makes them active in finding new pathways for electron hopping so that the chalcogen p orbitals can gain all electrons they need to form a closed shell system. Such pathways could, in principle, extend to a substrate. As an example, with Au(111) surface as a substrate, one can expect electron charge transfer from spatially extended 4s electrons of Au to the chalcogens above them. Meanwhile, the trigonal symmetry of Au(111) substrate enforces the original symmetry of pristine monolayer TaS 2 , thereby avoiding any lattice deformation. Of course, hybridization between monolayer TaS 2 and the Au(111) substrate could significantly affect its electronic band structure 38−40 , and correspondingly, its intrinsic properties. Alternatively, when the substrate is made of chemically inactive orbitals (such as p π states in graphene), the CDW is still inevitable.
The cited references are as follows, Inspired by the reviewer's comments and the works cited above, we think it will be interesting to further investigate the effects of the substrate on the CDW order in this family of materials. The data on the suspended thin flakes, shown in the original Suppelmentary Section 10, will form the basis of such a study. We are afraid that they will go largely unnoticed if they only appear in a supplementary file. If the reviewer thinks it is acceptable, we would like to remove these data from the Supplementary Information and leave them for a future publication. We believe this will serve as a better contribution to the field. Accordingly, we may change the following sentence in the Discussion section, "The exfoliated samples are expected to exhibit intrinsic CDW properties, verified by control experiments on suspended thin flakes (Supplementary Section 10).", with the content in the parenthesis replaced by "data to appear in a future publication". * It would be useful to compare these Raman spectroscopy results with those available in the recent literature, e.g. Phys. Rev. Lett. 122, 127001(2019), Phys. Rev. B 100, 165414 (2019), Phys. Rev. B 98, 165109 (2018) andPhys. Rev. B 97, 094502 (2018).
We appreciate the reviewer for suggesting to cite these works. They indeed help our discussion. Phys. Rev. Lett. 122, 127001 (2019) andPhys. Rev. B 97, 094502 (2018) indeed support our assignment of the CDW amplitude modes and zone-folded modes in 2H-TaS 2 and 2H-NbSe 2 , respectively. We add them as the new Refs. 30 and 31 in the revised version and cite them in the original Line 118.
Phys. Rev. B 100, 165414 (2019) is on the study of 1T-TaS 2 , which found a suppression of both the commensurate and the incommensurate CDW transitions in atomically thin sample. This is consistent with the results from an electrical transport study on the same system, as shown in the original Ref. 18. We therefore cite these two works together in the introduction.
Phys. Rev. B 98, 165109 (2018) mainly focuses on the layer-number and excitation energy dependence of the first-order phonons in NbSe 2 . The established layer-number dependence of the A 1g and E 2g phonon frequencies are consistent with ours shown in the Supplementary Information. We therefore cite this work for comparison.
In the revised manuscript, we have corrected these typos.
* Fig 1 (g): I would write the label as "bulk has harder/softer lattice", so that the panel can be understood without the caption. In the current version, it is not clear which situation has the harder lattice.
If the reviewer does not mind, we wish to keep them unchanged. The positive (negative) ∆Q σ region corresponds to a harder (softer) lattice as compared to a monolayer. This terminology is thus not limited to the bulk. It is applicable, for example, to the bilayer too, and in general any configuration beyond monolayer. That is why we simply write "harder lattice" and "softer lattice" and highlight each regime by different colours in the figure.
* Fig 1f and similar figures: It does not really make sense to connect the data points by lines here, and I find it a bit distracting.
We totally agree with the reviewer. Those line do not bear any physical meaning. In the revised manuscript, we have replaced the lines and dots in Figure 1f and 1g, Figure 2c and Figure 4b and represented our results with bar charts. * Line 92, "Pierels"→ "Peierls" Corrected in the revised manuscript.
As there are indeed partially contradictory theoretical predictions and also experimental observations reported in the literature on CDWs in this (and similar) type(s) of material, the present study makes a novel and valuable contribution to this field. The detailed Raman experiments seem to be carried out carefully (it is especially important that the bulk crystals were exfoliated in a glove box and subsequently encapsulated by h-BN), and also the data evaluation looks reasonable.
Furthermore, the three parameter-based model ( fig. 1a) proposed to account for the presence or absence (like in NbS 2 ) of CDW instability, and the thickness dependence of T CDW for the three different compounds, makes sense and is -despite its simplicity -in quite good agreement with the experimental observations. Although no fundamentally new theoretical result is provided, the presented model can nevertheless help to further clarify relevant issues on CDWs in 2D materials. Finally, also the mentioned experimental factors (in particular the role of the substrate) are indeed likely to have an influence on T CDW in these compounds.
We appreciate the reviewer's time to evaluate our work and are grateful for the positive comments regarding the novelty and significance of our results.