Possible phason-polaron effect on purely one dimensional charge order of Mo6Se6 nanowires

In one-dimensional (1D) metallic systems, the diverging electron susceptibility and electron-phonon coupling collaboratively drive the electrons into a charge density wave (CDW) state. However, strictly 1D system is unstable against perturbations, whose effect on CDW order requires clarification ideally with altered coupling to surroundings. Here, we fabricate such a system with nanowires of Mo6Se6 bundles, which are either attached to edges of monolayer MoSe2 or isolated freely, by post-annealing the preformed MoSe2. Using scanning tunneling microscopy (STM), we visualized charge modulations and CDW gaps with prominent coherent peaks in the edge-attached nanowires. Astonishingly, the CDW order becomes suppressed in the isolated nanowires, showing CDW correlation gaps without coherent peaks. The contrasting behavior, as revealed with theoretical modeling, is interpreted as the effect of phason-polarons on the 1D CDW state. Our work elucidates a possibly unprecedented many body effect that may be generic to strictly 1D system but undermined in quasi-1D system.


INTRODUCTION
When electrons are confined in an atomic chain, correlations among them get enhanced causing instability of the Fermi surface due to its perfect nesting and reduced screening 1 . This makes the systems susceptible to electron-electron and electron-phonon interactions, driving them into a rich variety of exotic correlated quantum states that are distinguished from 3D, such as the Tomonaga-Luttinger liquid, spin density waves, and CDW. In the CDW state, lattice symmetry is spontaneously broken via a Peierls distortion mechanism dictated by the electron-phonon coupling 2 . The electron density acquires a spatial modulation corresponding to twice the Fermi wave vector (kF), and concomitantly opens a gap around the Fermi energy ( ). However, quantum and thermal fluctuations in strictly 1D system induce uncertainty in nuclear positions of the same order as that of the Peierls distortion 3 , which tend to destroy the CDW coherence. Indeed, most experimental CDW orders are observed in quasi-1D systems [4][5][6] , that are ensembles of chains contained either in a bulk 3D or a surface 2D form.
Their inter-chain interaction is argued to stabilize the CDW state, but meanwhile causes complications. It is desirable to experimentally examine its existence in a truly 1D system.
Although recent experiments have reported CDW orders in single silicide nanowires 7 and mirror twin boundary of monolayer MoSe2 8 that are indeed electronically isolated 1D systems, they are still coupled to the lattice of supporting surroundings, whose phonons are 3 essentially 2D. Thus, a 1D system with altered coupling to its surroundings is essential to uncover the embedded properties in inherent CDW order. Creating such a system is challenging, because it should have 1D electronic character with reasonable regularity, defect-free from disorder and retain its 1D electronic property against altered coupling to the surroundings. Our strategy is to use transition metal monochalcogenide (TMM) nanowires, which have monolayer thickness, nanometer width and show metallic conductivity [9][10][11][12] . TMM is a polymorph of transition metal dichalcogenides (TMDs), which are widely studied due to their direct-band gaps with spin and valley polarization 13 at the single layer limit. Because of the polymorph, TMM nanowires can be obtained by judiciously tuning the stoichiometry and even achieving contacts to TMDs.
Here, we present an approach for fabricating TMM Mo6Se6 nanowires by postannealing the preformed TMD MoSe2 monolayers prepared with molecular beam epitaxy 21 on a graphene-covered SiC(0001) substrate 22 , producing straight nanowires with monolayer height and well-defined width. The nanowires are either attached to the MoSe2 edges or isolated freely, allowing altered coupling to the environment. Equally importantly, the graphene substrate has a negligible interaction with the supported nanowires. Both conditions promise the system ideal for the study of purely 1D CDW order. There is a Moiré period ~1.3 nm along the nanowires, implying their interface with MoSe2 is atomically smooth, and a larger one ~5 nm that is likely from imperfect Moiré overlapping ( Fig. S4A and S5).

Morphology
Next, we show their tunneling spectra. While MoSe2 has a band gap of ~2.2 eV 23 , Mo6Se6 exhibits increased conductance with multiple peaks below , a relatively flat but finite conductance above , and an enhanced peak at ~1.6 eV with a splitting of ~0.2 eV (Fig. 1C, Fig. S6). The edge-attached and isolated nanowires exhibit similar spectra, indicating their electronic structure of 1D nature is well-conserved against the edgeattachment to MoSe2 layer. Those spectroscopic features are captured by our DFT calculations of both a single Mo6Se6 wire (Fig. 1D) and a bundle containing 2-5 wires (3wire bundle exemplified in Fig. 1E), demonstrating their bands are not substantially modified by the bundled structure. Comparison with the experiment reveals the nanowires are electrondoped ~0.3 eV by the substrate (Fig. 1C) resulting in a F k~5.35 nm -1 (Fig. 1E).

CDW of Mo6Se6 nanowires
Intriguingly, spectra of a typical edge-attached nanowire ( Fig The observed gap cannot be stem from disorder-induced electron localization 24,25 , because it has a U-shape with coherence peaks and a spatially identical size throughout each nanowire. To explore its CDW origin, we examine the real-space conductance of the nanowire around energies of the coherence peaks. There it exhibits periodic modulations of 0.64 ± 0.04 nm, which coincide with the calculated CDW period of The CDW order in edge-attached nanowires has several properties. First, the CDW modulation is found incommensurate with the lattice and barely influenced by the strain- 6 releasing defects (Fig. S4E). Second, there exists conductance inhomogeneity along the nanowire that correlates with the imperfect Moiré overlapping. The spatial scale is consistent with the CDW correlation length of F 2 v  ~2.5 nm 26 (Fig. 2B, Fig. S4B). Third, a series of satellite peaks of ~15 ± 2 meV spacing appear next to the coherence peaks ( Interestingly, the isolated nanowire also exhibits a spectroscopic gap around at 4.4 K (Fig. 3, A and B), which features enhanced conductance, related to an adjacent electronic state, below the lower gap edge, and power-law shaped onset of the conductance at the upper gap edge. While the gap shape shows some difference at each spectrum along the nanowire, they all have similar gap size (~130 meV) except at both end-contacts, and surprisingly no coherence peaks (Fig. 3C). There is no charge modulation along the nanowire either (Fig.   3B). Similar spectroscopic gap has been observed in 10 isolated nanowires of different length.
The gap is off-centered from that may appear at either side (Fig. S10), and gets smeared intrinsically with increasing temperature (Fig. 3D

Phason-polaron model on CDW of different dimensionality
The above perturbative analysis gives weak influence of quantum fluctuations from acoustic phonons and phason on the CDW, which is inconsistent with the experimental 8 observation. This suggests the quantum fluctuation from the above aspects is possibly strong, which leads us to consider a polaronic scenario. When an electron is injected from or to the STM tip, it may shake up the phonons or phasons, resulting in a polaronic effect (Fig. 4, A and B). We have estimated the polaron coupling constant to the acoustic phonon 33  temperature effect, which in 2D case gives A ω ∝ ω Δ ℏ / with being the Boltzmann constant. Therefore, it is expected that the singularity of A ω disappears when ℏ / 4 giving ~36 K. The temperature dependence of A ω , which is quantified by the ratio of intensity between the coherence peak and the background, seems consistent with this expectation (Fig. 4E), giving another support for our picture. Note that our model is based on gapless phasons and assumes that the wires are infinitely long, which are not ideal in experiment. CDW pinning by disorder or commensurate lattice can introduce a phason gap 2 . Moreover, the finite length of the isolated wire introduces a crossover energy scale between the 1D and 2D phason. We have evaluated that their influence on phasons is negligible (Supplementary Note 5).

Outlook
Our study envisions several future studies. The phason-polaron effect suggested here should be generic to strictly 1D CDW states, which can be examined in more experimental systems. Mo6Se6 bundles with increased width, on the other hand, are expected to induce a dimensional crossover from strictly 1D to quasi-1D, whose influence on the CDW states can be studied systematically. Moreover, the polaronic effect with different dimensionality may also interplay with other correlated states in 1D system, such as spin density waves, Tomonaga-Luttinger liquid 34 , etc., which opens up new directions for in-depth investigations.

Sample preparation.
SiC (0001)  Further extended annealing degrades partial MoSe2 layers, and meanwhile forms some isolated Mo6Se6 nanowires connecting to MoSe2 islands at two ends (Fig. S3D). These wires are tantalizing for making interconnects to the monolayer circuits.

STM measurement.
The experiments were performed with a cryogenic custom-made Unisoku STM 35 . An electrochemically etched W wire was used as the STM tip. Prior to measurements, the tip was characterized on a Ag(111) multi-layer films grown on a Si(111) substrate, which has been cleaned by several flashing cycles to 1500 K. The tunneling spectra were obtained by lockin detection of the tunneling current with a modulation voltage at 983 Hz feeding into the sample bias.

DFT calculation.
The electronic structure and the DOS of Mo6Se6 nanowires were calculated in the framework of DFT within the generalized gradient approximation 36  and a 3-wire bundle (E), whose crystal structure is same as that in (B). The solid and dashed green lines in (C to E) mark of the measured and calculated nanowires, respectively. The black circle marks F k . The DOS of (E) is shown in (C) for comparison.