Quantum enhancement of charge density wave in NbS$_2$ in the 2D limit

TL;DR

This study uses first-principles anharmonic calculations to reveal how quantum effects enhance charge density wave formation in monolayer NbS2, contrasting with bulk behavior and suggesting potential for tunable electronic devices.

Contribution

It provides the first detailed quantum anharmonic analysis of charge density wave phenomena in NbS2, explaining experimental discrepancies and predicting strain-induced switching capabilities.

Findings

Bulk NbS2 shows no charge order due to anharmonicity.

Monolayer NbS2 exhibits a $3\times 3$ charge density wave reconstruction.

Small strains can switch charge order on and off in monolayer NbS2.

Abstract

At ambient pressure, bulk 2H-NbS$_2$ displays no charge density wave instability at odds with the isostructural and isoelectronic compounds 2H-NbSe$_2$, 2H-TaS$_2$ and 2H-TaSe$_2$, and in disagreement with harmonic calculations. Contradictory experimental results have been reported in supported single layers, as 1H-NbS$_2$ on Au(111) does not display a charge density wave, while 1H-NbS$_2$ on 6H-SiC(0001) endures a $3\times 3$ reconstruction. Here, by carrying out quantum anharmonic calculations from first-principles, we evaluate the temperature dependence of phonon spectra in NbS$_2$ bulk and single layer as a function of pressure/strain. For bulk 2H-NbS$_2$, we find excellent agreement with inelastic X-ray spectra and demonstrate the removal of charge ordering due to anharmonicity. In the 2D limit, we find an enhanced tendency toward charge density wave order. Freestanding 1H-NbS$_2$ undergoes a $3\times3$ reconstruction, in agreement with data on 6H-SiC(0001) supported samples. Moreover, as strains smaller than $0.5\%$ in the lattice parameter are enough to completely remove the $3\times3$ superstructure, deposition of 1H-NbS$_2$ on flexible substrates or a small charge transfer via field-effect could lead to devices with dynamical switching on/off of charge order.

Full text

Transition metal dichalcogenides (TMDs) are layered materials with generic formula MX2, where M is a transition metal (Nb, Ta, Ti, Mo, W, …) and X a chalcogen (S, Se, Te). The layers, made of triangular lattices of transition metal atoms sandwiched by covalently bonded chalcogens, are held together by weak van der Waals forces, and TMDs can be readily exfoliated into thin flakes down to the single layer limit, with mechanical or chemical techniques Novoselov et al. (2005); Mak et al. (2012); Radisavljevic et al. (2011); Zeng et al. (2011). In TMDs, the interplay between strong electron-electron and electron-phonon interactions gives rise to rich phase diagrams, with a wide variety of cooperating/competing collective electronic orderings as charge-density wave (CDW), Mott insulating, and superconductive phases Wilson et al. (1975); Calandra (2018). Of the several polytypes, we focus here on the most common one for NbS2, the H polytype Leroux et al. (2012, 2018), where the transition metal is in trigonal prismatic coordination with the surrounding chalcogens. In Fig. 1 the 1H (monolayer) and 2H (bulk) crystal structures are shown.

Among metallic 2H bulk TMDs, NbS2 occupies a special place since no CDW has been reported Fisher and Sienko (1980); not , contrary to its isoelectronic and isostructural 2H-TaSe2, 2H-TaS2 and 2H-NbSe2. All these systems have very similar band structures and are conventional (i.e. phonon-mediated) superconductors with critical temperatures $T_{c}$ that increases from a sub-Kelvin value in 2H-TaSe2 and 2H-TaS2 (around $0.2$ K and $0.5$ K, respectively) up to $5.7$ K in 2H-NbS2 and $7.2$ K in 2H-NbSe2 Navarro-Moratalla et al. (2016); Wagner et al. (2008); Harper et al. (1977); Heil et al. (2017). They also show quite a different CDW transition strength Naito and Tanaka (1982); Castro Neto (2001). 2H-TaSe2, 2H-TaS2 and 2H-NbSe2 undergo a triple incommensurate CDW transition to a superlattice with hexagonal symmetry corresponding roughly to the same wave-vector $\boldsymbol{q}_{\scriptscriptstyle{\text{CDW}}}=\Gamma\,M(1-\delta)2/3$ ($\delta\simeq 0.02$ is the incommensurate factor) of the Brillouin zone. However, the transition temperature $T_{\scriptscriptstyle{\text{CDW}}}$ increases from $30$ K for 2H-NbSe2 to $80$ K for 2H-TaS2 and $120$ K for 2H-TaSe2 (2H-TaSe2 actually shows a further commensurate first-order CDW transition at $92$ K with $\delta$ dropping continuously to zero) Moncton et al. (1977). Therefore, 2H-NbS2 considerably stands out as it shows only an incipient instability near $\boldsymbol{q}_{\scriptscriptstyle{\text{CDW}}}$, but it remains stable even at the lowest temperatures. This circumstance is even more surprising if it is considered that 2H-NbSe2 and 2H-NbS2 display superconductivity at similar temperatures.

In TMDs, the behavior of the CDW ordering in the two-dimensional limit cannot be inferred from the knowledge of their bulk counterparts, since two competing mechanisms are expected to play a major role. On the one hand, reduced dimensionality strengthens Peierls instabilities (due to Fermi surface nesting) and electron-phonon interactions (due to reduced dielectric screening), thus favoring stronger CDW. On the other hand, stronger fluctuation effects from both finite temperatures and disorders should tend to destroy long-range CDW coherence in low-dimensional systems Xi et al. (2015). In particular, the effect of dimensionality on the CDW ordering in the H polytype is a current active research area. In 1H-TaS2, the CDW vanishes in the 2D limit Yang et al. (2018), while in 1H-TaSe2 it remains unchanged with respect to the bulk Ryu et al. (2018). For 1H-NbSe2 and 1H-NbS2 the situation is more debated. In the 1H-NbSe2 case, $3\times 3$ CDW is observed, but some controversy is still present in literature, tentatively attributed either to the sample exposure to air or to the different substrates, concerning the thickness dependence of the $T_{\scriptscriptstyle{\text{CDW}}}$ (lower/higher $T_{\scriptscriptstyle{\text{CDW}}}$ of the monolayer with respect to the bulk has been reported with bilayer graphene Ugeda et al. (2015)/silicon Xi et al. (2015) substrate, respectively). Supported single layers of 1H-NbS2 have become recently available, and while no traces of CDW have been observed down to $30$ K for monolayers grown on top of Au(111) Stan et al. (2019), a $3\times 3$ CDW ordering has been observed at ultra-low temperature (measurements performed below $5$ K) for monolayers grown on top of graphitized 6H-SiC(0001) Lin et al. (2018).

In this letter we investigate, from first-principles, the vibrational properties of bulk 2H-NbS2 (at zero and finite pressure) and suspended 1H-NbS2, taking into account quantum anharmonic effects at non-perturbative level in the framework of the stochastic self-consistent harmonic approximation (SSCHA)  Errea et al. (2014); Bianco et al. (2017); Monacelli et al. (2018); not . For bulk 2H-NbS2, we show that quantum anharmonic effects remove the instability found at harmonic level, and give temperature dependent phonon energies in quantitative agreement with experiment. Previous anharmonic calculations for 2H-NbS2 anticipated the role of anharmonicity, but were limited to a low dimensional subspace of the total high dimensional configurations space and did not account for the temperature dependence Heil et al. (2017). We also show that quantum anharmonic effects are noticeable even at high pressure. Moreover, we demonstrate that the difference between 2H-NbS2 and 2H-NbSe2 is not simply ascribable to the different chalcogen mass. Finally, we analyze the 2D limit and show that freestanding single-layer 1H-NbS2 undergoes a $3\times 3$ CDW instability in agreement with data on 6H-SiC(0001) supported samples. However, strains smaller than $0.5\%$ are sufficient to completely remove the instability, suggesting a strong dependence of the CDW on the environmental conditions (substrate, charge transfer…) and reconciling the apparent contradiction with supported Au(111) samples.

For bulk 2H-NbS2, in Fig. 2 we compare the computed anharmonic phonon dispersions with the results of the inelastic X-ray scattering (IXS) experiment of Ref. 7, at low and ambient temperature. We also show the (temperature-independent) harmonic phonon dispersion. Calculations were performed with the $a^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Exp}}}=b^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Exp}}}=3.33\,\textup{\AA}$ and $c^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Exp}}}=11.95\,\textup{\AA}$ bulk experimental lattice parameters at zero pressure Leroux et al. (2012). The phonon dispersion is almost everywhere well reproduced with the harmonic calculation, except close to $M$, where it predicts that two longitudinal acoustic and optical modes become imaginary. Experimental phonon energies show a sensible temperature dependence in this region of the $BZ$ and are, obviously, always real. The SSCHA cures the pathology of the harmonic result: the anharmonic phonon dispersions do not show any instability and give a very good agreement with the experiment at both temperatures.

Since SSCHA calculations give dispersions in good agreement with experiments, we can perform a wider analysis. In the upper panel of Fig. 3 we show the SSCHA phonon dispersion for different temperatures along the full high-symmetry path of the $BZ$. As temperature decreases, anharmonicity causes the softening of two acoustic and optical longitudinal modes close to both $M$ and $L$, but there is no instability. Thus, quantum fluctuations strongly affected by the anharmonic potential stabilize 2H-NbS2. In the other two panels we show the effect of hydrostatic pressure on the phonon dispersion. Since there are no available experimental lattice parameters at high pressures, we estimated them by assuming that the ratio between experimental and standard DFT theoretical lattice parameters (i.e. the lattice parameters that minimize the DFT energy but do not take into account any lattice quantum dynamic effects), $a^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Exp}}}(P)/a^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Th-DFT}}}(P)$ and $c^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Exp}}}(P)/c^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Th-DFT}}}(P)$, are independent of the applied pressure $P$. Thus we computed those ratios at zero pressure and, for a given pressure $P$, the calculations were performed using as lattice parameters $a=\left(a^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Exp}}}/a^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Th-DFT}}}\right)\times a^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Th-DFT}}}(P)$ and $c=\left(c^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Exp}}}/c^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Th-DFT}}}\right)\times c^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Th-DFT}}}(P)$. Increasing pressure the anharmonicity of the lowest energy modes around $M$ and $L$ decreases, but remains relevant even up to 14 GPa. A similar conclusion was drawn for 2H-NbSe2, where large anharmonic effects and strong temperature dependence of these phonon modes were observed as high as 16 GPa, in a region of its phase diagram where no CDW transition is observed Leroux et al. (2015).

These results confirm the importance of quantum anharmonicity in 2H-NbS2 to describe experimental data and the absence of a CDW instability. It is tempting, at this point, to use the same technique to shed light on the different CDW behavior exhibited by the very similar compound 2H-NbSe2. Indeed, as we showed in a previous work Leroux et al. (2015), the SSCHA correctly displays the occurrence of CDW in 2H-NbSe2 at ambient pressure. One evident difference between 2H-NbS2 and 2H-NbSe2 is, of course, the mass of the chalcogen atom. We then performed a SSCHA calculation at [math] K for 2H-NbS2 with “artificial” S atoms having unaltered electronic configuration but the mass of Se. In other words, we performed a SSCHA calculation where the average displacements of the atoms from the equilibrium position is ruled by the Se mass, but for each fixed position of the atoms the electronic structure is computed with the normal S atoms. The results are shown in Fig. 4. Also in this case, when quantum anharmonic effects are included the system does not show any CDW instability. Thus the different behavior of 2H-NbS2 and 2H-NbSe2 cannot be ascribed to a mass effect but has a more complex origin related to the different electron screening on the ions.

The validity of the results obtained with the SSCHA method on bulk 2H-NbS2 gives us confidence that a similar calculation on the 1H-NbS2 monolayer may shine light about the effects that dimensionality and environmental conditions (substrate, doping) can have on the CDW ordering in metallic TMDs. The suspended 1H-NbS2 monolayer was simulated leaving $12.55\,\textup{\AA}$ of vacuum space between a 1H layer and its periodic replica. At conventional static DFT level, we found that the theoretical zero pressure in-plane lattice parameter of the monolayer and the bulk are essentially the same, $a^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Th-DFT}}}\simeq a^{\scriptscriptstyle{\text{1H}}}_{\scriptscriptstyle{\text{Th-DFT}}}\simeq 3.34\textup{\AA}$. Therefore, for the suspended monolayer we use as in-plane lattice parameter the bulk experimental one, $a^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Exp}}}=3.33\,\textup{\AA}$. This value is also compatible with the recent experimental measures $3.29\pm 0.03\,\textup{\AA}$ and $3.34\,\textup{\AA}$ reported for the lattice parameter of monolayer grown on substrate in Ref. 23 and Ref. 22, respectively.

In the upper panel of Fig. 5, we show the harmonic and SSCHA anharmonic phonon dispersions of suspended 1H-NbS2 at several temperatures, calculated with the lattice parameter $a^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Exp}}}$. As in the bulk case, the system is unstable at harmonic level, but it is stabilized by quantum fluctuations strongly sensitive to the anharmonic potential down to [math] K. However, comparing Figs. 2 and 5, we observe that even if the used in-plane lattice parameter is the same in both cases, at [math] K the softest theoretical phonon frequency is approximately $20\%$ harder in the bulk than in the single layer case, demonstrating that there is a substantial enhancement of the tendency toward CDW in the 2D limit. In the monolayer, the theoretical phonon softening is localized in $\boldsymbol{q}_{\scriptscriptstyle{\text{CDW}}}=0.72\,\overline{\Gamma}\overline{M}$, which is quite close to the $\boldsymbol{q}_{\scriptscriptstyle{\text{CDW}}}\simeq 2/3\,\overline{\Gamma}\overline{M}$ of the CDW instability experimentally found in 1H-NbS2 on 6H-SiC(0001) Lin et al. (2018) (and in 1H-NbSe2 Xi et al. (2015); Ugeda et al. (2015)). Notice that, since the computed wave-vector of the instability may be affected by the finite grids used in the calculations, we do not discard that it may be slightly shifted in the infinite grid limit.

Pressure tends normally to remove CDW ordering. Therefore, considering the proximity of the instability, it cannot be discarded that a tensile dilatation due to the substrate may induce the CDW transition observed for 1H-NbS2 on graphitized 6H-SiC(0001). However, for the same reason, we cannot exclude the more interesting prospect that the observed CDW be an intrinsic property of this system. Indeed, even small variations of the lattice parameter, compatible with the experimental uncertainly, could have a relevant impact on the results of the calculations, and a more accurate theoretical analysis of the monolayer structure is therefore necessary. As the energy of the soft-mode along $\overline{\Gamma M}$ is of the order of $\simeq 58$ K, for a proper analysis of the CDW in the monolayer it is important to fully take into account quantum effects. Including quantum anharmonic contributions to strain through the technique introduced in Ref. 26, we find that with the used lattice parameter $a^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Exp}}}$ the structure is sligthly compressed, with an in-plane pressure $P=0.66$ GPa. Upon relaxation we obtain the theoretical lattice parameter $a^{\scriptscriptstyle{\text{1H}}}_{\scriptscriptstyle{\text{Th}}}=3.35\,\textup{\AA}$, approximatively $0.5\%$ larger than $a^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Exp}}}$.

The harmonic and quantum anharmonic phonons at [math] K calculated with the lattice parameter $a^{\scriptscriptstyle{\text{1H}}}_{\scriptscriptstyle{\text{Th}}}$ are shown in the bottom panel of Fig. 5. While at harmonic level the phonon dispersion is not substantially different from the one computed with $a^{\scriptscriptstyle{\text{2H}}}_{\scriptscriptstyle{\text{Exp}}}$, when quantum anharmonic effects are included the phonon dispersion at [math] K now shows an instability at $\boldsymbol{q}_{\scriptscriptstyle{\text{CDW}}}=0.72\,\overline{\Gamma}\overline{M}$, thus in agreement with the CDW observed for 1H-NbS2 on top of 6H-SiC(0001). The obtained instability is very weak (i.e. the obtained imaginary phonon frequency is very small). Therefore, this result is also compatible with the hypothesis that charge doping from the substrate could be at the origin of the CDW suppression for 1H-NbS2 on top of Au(111), similarly to what it was proposed for the case of 1H-TaS2 on top of Au(111) Albertini et al. (2017). Our results show that if quantum anharmonic effects are included, then even a small compression/dilatation of approximately $0.5\%$ removes/induces the charge density wave instability on 1H-NbS2. The extreme sensitivity of the CDW on environmental conditions therefore suggests that deposition of 1H-NbS2 on flexible substrates Wu et al. (2015); He et al. (2013); Conley et al. (2013), or a small charge transfer via field effect, could lead to devices with dynamical on/off switching of the $3\times 3$ order.

In conclusion, we have shown that quantum anharmonicity is the key interaction for the stabilization of the crystal lattice in bulk 2H-NbS2, as it removes the instability found at the harmonic level. The calculated temperature dependence of the phonon spectra are in excellent agreement with inelastic X-ray scattering data. Anharmonicity remains important even at large pressures. Given the good agreement between theory and experiment in bulk 2H-NbS2, we have studied the behavior of the CDW in the 2D limit by considering single layer 1H-NbS2. We found that suspended 1H-NbS2 undergoes a quantum phase transition to a CDW state with approximately $3\times 3$ charge ordering in the 2D limit, in agreement with experimental results on supported samples on 6H-SiC(0001). However, the CDW is extremely sensitive to environmental conditions, as it is very weak and compressive strains smaller than $0.5\%$ are enough to suppress it. This explains the absence of CDW observed in 1H-NbS2 on top of Au(111). This also suggest that devices with dynamical on/off switching of the $3\times 3$ charge order can be obtained with deposition of 1H-NbS2 on flexible substrates, or through a small charge transfer via field effect.

Acnowledgement

R.B. acknowledges the CINECA award under the ISCRA initiative (Grant HP10BLTB9A). Computational resources were provided by PRACE (Project No. 2017174186) and EDARI(Grant A0050901202). I.E. acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (Grant No. FIS2016-76617-P). M.C. acknowledges support from Agence Nationale de la Recherche under the reference No. ANR-13-IS10-0003- 01. We acknowledge support from the Graphene Flagship (Grant Agreement No. 696656-GrapheneCore1).