Slow dynamics of disordered zigzag chain molecules in layered LiVS2 under electron irradiation

Electronic instabilities in transition metal compounds often spontaneously form orbital molecules, which consist of orbital-coupled metal ions at low temperature. Recent local structural studies utilizing the pair distribution function revealed that preformed orbital molecules appear disordered even in the high-temperature paramagnetic phase. However, it is unclear whether preformed orbital molecules are dynamic or static. Here, we provide clear experimental evidence of the slow dynamics of disordered orbital molecules realized in the high-temperature paramagnetic phase of LiVS2, which exhibits vanadium trimerization upon cooling below 314 K. Unexpectedly, the preformed orbital molecules appear as a disordered zigzag chain that fluctuate in both time and space under electron irradiation. Our findings should advance studies on soft matter physics realized in an inorganic material due to disordered orbital molecules.

Kimber et al. predicted that a classical analogue of the short-range Resonating Valence Bond state will be realized in the high-temperature phase of Li 2 RuO 3 as the existing dimer patterns resonate due to thermal fluctuations 13 . They explained that slower dynamics should appear compared to the characteristic timescale of their x-ray measurements, although these dynamics have yet to be experimentally clarified. By contrast, Browne et al. performed a quasi-elastic neutron scattering experiment up to 1100 K in GaV 2 O 4 with disordered trimer and tetramer pairs in anticipation of the appearance of fast dynamics. However, they concluded that clusters are well-defined and statically disordered even at 1100 K as they were unable to detect dynamic behaviours faster than ~ 1×10 -11 s 12 .
Here, we present comprehensive structural studies of LiVS 2 , which exhibits a paramagnetic metal to nonmagnetic insulator transition at 314 K 5 . A recent structural study clarified that vanadium trimer molecules form in the low-temperature nonmagnetic phase 6 .
The vanadium trimer molecules disappear above 314 K, and unprecedented zigzag chain molecules consisting of dimers emerge with a finite correlation length. Cooling increases the correlation length, and sharp superstructure peaks grow in powder diffraction patterns below approximately 350 K.
Our annular dark-field scanning transmission electron microscope (ADF-STEM) experiment clearly shows that these existing zigzag chain molecules slowly fluctuate in both time and space on the order of seconds. The high-temperature phase with slow dynamics of some existing zigzag chain patterns is categorised as a plastic crystal phase, which is a mesophase between a crystal and liquid. The atoms retain the original position in the time average but the orientation of the zigzag chain fluctuates.

Experimental Results
Physical properties of LiVS 2 Figure 1a displays the physical properties of LiVS 2 . It exhibits a metal to nonmagnetic insulator transition around 314 K. As a previous study has already clarified 5 , the high-temperature paramagnetic susceptibility increases upon heating, which is reminiscent of pseudogap behavior found in underdoped high-T c cuprates. Although the present data are consistent with those reported previously, the present entropy change of ∆ = 7.99 J/mol K at 314 K is much larger than ∆ = 6.6 J/mol K in the previous study 5 . This difference may be due to the improved sample quality in this study. For checking the quality of the present sample, we performed the Inductively Coupled Plasma-Atomic Emission Spectrometry (ICP-AES) experiment, and confirmed that the Li content, x, is 0.97 (2) in Li x VS 2 . A subtle amount of off-stoichiometry does not have a significant effect on the physical properties, as mentioned above. Note that no discontinuous behaviors appear in these physical property data above 314 K for the discussion later. Figure 1b shows the synchrotron x-ray diffraction patterns as functions of temperature.

Rietveld analysis
Below 314 K, sharp superstructure peaks emerge around ~ 3.2-3.3 Å −1 . Recently, the low-temperature crystal structure was solved using a Rietveld method by assuming a trigonal space group 31 . It was clarified that the vanadium trimers in a spin singlet state spontaneously form (Figure 1c) 6 .
Upon heating above 314 K, the superstructure peaks associated with vanadium trimerization completely disappear in the powder diffraction data and unprecedented additional superstructure peaks emerge (Figure 1b, arrows), which has never been reported.
The additional peaks gradually weaken upon further heating and eventually disappear at approximately 350 K. For the higher temperature data, refinement can be successfully performed by assuming a trigonal space group 3 ̅ 1 with a regular triangular lattice, while the refinement can be successfully performed for powder diffraction data obtained at 320 K by assuming a monoclinic space group with vanadium zigzag chain molecules ( Figure   1d). Strangely, the distinct superstructure peaks clearly show the monoclinic displacement of the consisting atoms, the monoclinic lattice strain from the triangular lattice is negligible.
Both the temperature dependent lattice parameters and the details of the Rietveld refinement are summarized in the Supplemental Information. Figure 1e-g display the high-resolution x-ray diffraction patterns at 320 K with the indices. The emergences of the superstructure peaks with ≠ 0 in the indices indicate that the zigzag chain appears as three-dimensional ordering.

PDF analysis
Pair Distribution Function (PDF) analysis was performed to evaluate whether the preformed orbital molecules appear in a disordered form in the high-temperature paramagnetic phase. Figure 2a  x-ray diffraction patterns at 320 K with indices of monoclinic phase. f-g, Enlarged views of a fundamental 001 peak (f) and a superstructure 103 peak (g) with the correlation length estimated using the Scherrer equation. Instrumental resolution is not taken into account. These data are obtained from the high-resolution data. simulation generates an unrealized peak at ~ 3.0 Å, which corresponds to the inner-trimer V-V distance. This clearly indicates that disordered vanadium trimers are absent in the high-temperature paramagnetic phase. Secondly, the fit assuming the average 3 ̅ 1 model with the regular vanadium lattice is also poor ( = 14.6%). The peak at ~ 3.4 Å, which corresponds to adjacent V-V and S-S distances, is much broader than the peak at ~ 2.5 Å, which consists of the adjacent V-S distance. We cannot successfully fit both peaks by assuming the average 3 ̅ 1 model even if we assume anomalous thermal parameters for V and S ions. If the average 3 ̅ 1 model is realized, the peak at ~ 3.4 Å should be stronger and sharper. Therefore, we can also exclude the regular triangle from the candidates. The experiment confirmed that the monoclinic crystal structure with a vanadium zigzag chain gives the best fit (

STEM experiment
The strong temperature dependence on correlation length expects us that the zigzag chain molecules have dynamic properties. To clarify whether the preformed zigzag chain molecules are dynamic or static, we observed a time series of high-resolution annular dark-field scanning transmission electron microscope (ADF-STEM) images of LiVS 2 at room temperature. In the thick part of the piece away from the edge, such as in the lower left portion of the inset of Figure 3a, the diffraction pattern derived from the 31 phase having  Another possibility within this scenario is that the Li at the edges is strongly diffused at room temperature due to the thermal and/or electron irradiation effects. A third possible scenario is that the phase transition temperature can change with sample thickness, as is often the case with some transition metal dichalcogenides 18,19 . For example, in 1T-TaS 2 , the low temperature Commensurate Charge Density Wave (CCDW) phase is suppressed when the sample thickness is below the threshold of ~ 40 nm 19 . Layer thicknesses at the edge of current samples are expected to be up to a few nanometers since it is not possible to perform ADF-STEM experiments on thicker samples. A fourth possible scenario is that the anomalous charge-fluctuating phase accompanied by lattice dynamics is induced by electron irradiation. A reference phenomenon has been observed in Ba 3 NaRu 2 O 9 , which contains the Ru 2 O 9 dimer, where the charge order melts under photoexcitation 20 . Further experimental studies should be required to clarify which scenario is realized in the future.

Theoretical considerations
After identifying the unprecedented dynamics of disordered zigzag chain molecules appearing in the high-temperature paramagnetic phase of LiVS 2 , we considered the underlying physics generating such dynamics of disordered zigzag chain molecules. Figure   4a shows the band calculation result based on the parameters obtained from the Rietveld analysis of 360 K data by assuming the trigonal space group 3 ̅ 1. The triply degenerate 2 orbitals, which consist of , , and orbitals, are oriented toward the neighbouring vanadium ions and are inherently separated into lower doubly degenerate ′ orbitals and a higher nondegenerate 1 orbital due to the trigonal splitting ( Figure 4d).
The corresponding first principles calculation result for a 3 ̅ 1 structure with = 4 eV suggests a Fermi surface with the possible common nesting vector of = * /2 ( Figure   4e), which can cause multiple dimerizations of the lattice along the -and ( + )-direction.
As shown in Figure 4d, this leads to the zigzag chain pattern. Our band calculation result obtained using the parameters of the local structure at 360 K clearly shows that the + component, which is due to the doubly degenerate ′ bands in the high-temperature 3 ̅ 1 phase, is separated into bonding and antibonding bands.  (Figure 4c). We speculate that the transition at 314 K induces a large band energy at the expense of lattice energy.

Discussion
As discussed above, the short-range order of the zigzag chains appearing in the high temperature phase of LiVS 2 can be originating from the orbitally assisted CDW instability. C. Conventionally, plastic crystals are realized in weakly interacting molecules or ions, where the consisting molecules/ions are thermally rotating at a fixed position. The dynamics of conventional plastic crystals are usually studied via NMR techniques. Considering that NMR covers the kHz-MHz order, we can estimate that the zigzag chain dynamics realized in the present LiVS 2 should be some order of magnitude slower than the rotation dynamics observed in conventional plastic crystals. We speculate that this is due to the strong interaction among neighbouring atoms, which depend on the complex network structure of inorganic materials. Our findings should provide a new platform for investigating soft matter physics in inorganic materials as well as expand the fundamental understanding of systems with preformed orbital molecules at high temperature.
In summary, we first observed the slow dynamics of disordered orbital molecules appearing in the high-temperature paramagnetic phase of LiVS 2 . The unconventional coupling between orbital and phonon should be an ingredient for generating the unprecedented dynamics, surely impact the studies of conventional orbital physics, such as nematic state in iron selenides [24][25][26] . Motivated by this work, we expect that further explorations of dynamics targeting similar systems, such as Li 2 RuO 3 and AlV 2 O 4 , should be accelerated, leading to the novel research fields of disordered orbital molecules.

Sample growth and preparation
Powder samples of LiVS 2 were prepared by a soft-chemical method and a subsequent solid-state reaction. Li-deficient Li ∼0.75 VS 2 was synthesised initially by a reaction with an appropriate amount of Li 2 S, V, and S in an Ar-filled quartz tube at 700 ℃ for 3 days. Note that the obtained precursors include small single crystalline samples with the order of m.
The products were put in 0.2 M n-BuLi hexane solution for 2 days to attain the maximum Li content 27 . The Li content, x, was confirmed to be 0.97 (2) in Li x VS2 from the Inductively

Physical property measurements
Differential scanning calorimetry (DSC) was conducted using a DSC 204 F1 Phoenix (Netzsch). The magnetic susceptibility was measured by a SQUID magnetometer (Quantum Design). The electrical resistivity was measured by the four-probe method. The powder samples were sintered at a low temperature of 300 ℃ because the inserted Li ions were partially deintercalated at higher temperatures. Experiments were performed in an Ar atmosphere.

Powder diffraction experiments
To investigate the average structure, synchrotron powder

Computational details
We employed the WIEN2k code 32  From the Bragg peak positions, we can easily find the primitive monoclinic unit cell with monoclinic = trigonal × √3 is realized at 320 K, as shown in Supplementary Figure 1 left, although the monoclinic distortion can be hardly found. Furthermore, we can also find extinction rule does not exist at 320 K diffraction patterns, resulting that the possible space group can be limited to 2, or 2/ . After some Rietveld analysis trials, it was clarified that no significant atomic displacement due to the two-fold symmetry occurred, but only the atomic displacement derived from the mirror symmetry appeared. Therefore, we can uniquely determine the space group to be at 320 K.
While a big jump appears corresponding to the trimer transition at around 314 K, the lattice parameters change smoothly across the transition from 3 ̅ 1 to at around 350 K. This is consistent with the fact that the transition happens on the increasing process of the correlation length. Correspondingly, the angle retains almost 90 degrees despite of the averaged monoclinic structure. Note that the β angle (= 90 degree) of the 3 ̅ 1 phase is defined by assuming a monoclinic unit cell.
Supplementary Figure 2 shows isotropic atomic displacement parameters ( ) of sulfur and vanadium, obtained from the Rietveld experiment of powder diffraction data. For refinements, a 31 structure (trimer structure) is assumed for low temperature data below 314 K, while a 3 ̅ 1 structure (regular triangle structure) is assumed for all high temperature data above 314 K. Although the superstructure peaks appear between 314 K and 350 K, trigonal space group of was included as an impurity phase, we successfully co-refined with a main phase of LiVS 2 .
The lattice parameters for Li deficient phase was refined from the initial parameters of 300 K data. The insets show the expanded data on the logarithmic scale. In the Rietveld analysis, we fixed one vanadium site to be (0, 0, 0). This is because we need to fix x and z on one site when the space group is assumed, which is consisting of three atomic sites: 2c (x,y,z), 1b (x, 1/2, z) and 1a (x,0,z). Supplementary Figure 9. Enlarged peaks for 400 K (P-3m1 phase) data with a Rietveld fitting. HWHM are almost consistent between these peaks.

Supplementary Note 6 : Q[S(Q)-1] vs Q plot
Here we show the Q dependences on Q[S(Q)-1] data at various temperature ranges. The data was used to obtain the reduced G(r) data for PDF analysis. This plot allows one to see if the diffraction signal is anomalously damped due to short range disorder, and also to see what short-range order is left at high-Q. As shown in Supplementary Figure 11, we can clearly find the Q[S(Q)-1] data at 325 K and 400 K show similar Q dependencies, which support that the similar short-range ordering is realized at 325 K and 400 K. These data are quite different from 300 K data.

Supplementary Note 7 : Estimation of the correlation length.
Just above the T trimer , the correlation length is long enough to generate sharp superstructure peaks in powder diffraction experiments, indicating zigzag chain formation.
The widths of the superstructure peaks are almost comparable to those of Bragg peaks, as shown in Figure 1e-g in the main text. Although the accurate estimation of correlation length based on the width of superstructure peaks cannot be performed because the width of superstructure peaks are almost limited by the instrumental resolution, it should be noted that the correlation length roughly estimated by applying Scherrer's equation to 103 peak of structure at 320 K, which appears at around Q ~ 3.26 as shown in Figure 2g, reaches to ~ 2200 Å.
Supplementary Figure 12 displays the reliable factor, , obtained by refining the data obtained at 325 K and 370 K by assuming the structure for various ranges. We can clearly find that is almost consistent between the data at 325 K and 370 K. This again indicates that the correlation length is longer than ~ 100 Å even at 370 K. We cannot perform the PDF analysis for higher regions because the reduced PDF G(r) becomes too noisy for performing the reliable PDF analysis.
Supplementary Figure 12 : The reliable factor, , as a function of . The refinement was performed in the range of − 10 ≤ ≤ . Here, we present the synchrotron x-ray diffraction patterns of Li deficient samples. Li deficient samples were prepared by the soft chemical technique using I 2 acetonitrile solution S1 . We can find that the superstructure peaks of 3 ̅ 1 phase are robust even at 300 K in x = 0.91(1), while the peaks disappear at around 250 K in x = 0.78 (1). Note that the Li content was determined using the lattice parameter at 400 K. It is known that the lattice parameter linearly changes depending on the composition from x = 0 to x = 1 at 400 K S2 .
The incident x-ray energy of E = 30 keV was used. # indicates the impurity of Li 2 S.

Supplementary Note 9 : Time-series of ADF-STEM experiment
A movie file of the time series images (STEM_dynamics_movie.gif) is available as an attached file of supplemental information. Movie legend is shown below.

Supplementary Note 10 : Phonon dispersion in the high temperature phase investigated using the first-principles calculation
To directly investigate the instability of the crystal structure of the LiVS 2 with a trigonal space group 3 ̅ 1, we performed the first-principles calculation of phonon dispersions, using the QuantumESPRESSO package 27,28 with the revised Perdew-Burke-Ernzerhof generalized gradient approximation 29 and the projector augmented-wave pseudopotential by Kresse and Joubert 30,31 . The plane-wave cut-off energy was set to 150 Ry, and the k-point mesh on the 12 × 12 × 6 Monkhorst-Pack grid 32 was used. The Methfessel-Paxton scheme 33 with a smearing width of 0.01 Ry was applied since the electronic structure is metallic. The cell structure was fully relaxed before calculating the dynamical matrix, and the interatomic force constants were obtained via the density functional perturbation theory (DFPT) 34 on the 4 × 4 × 2 grid.
The phonon dispersion of LiVS 2 with a trigonal space group 3 ̅ 1 is illustrated in Supplementary Figure 14. The acoustic phonon branch shows a negative-value frequency, implying that the crystal structure used in the calculation is unstable. The most negative value of the phonon frequency appears at the M point, and the corresponding wave vector coincides with the nesting vector calculated in the main text. We also confirm that the associated eigenvectors at the M point give the zigzag-chain-type atomic displacements (Supplemental data, M.gif) S3 . Therefore, the analysis by DFPT also supports the fact that LiVS 2 with a trigonal space group 3 ̅ 1 has the instability to form the zigzag chain.
Supplementary Movie I : Time series images of Fourier-masked ADF-STEM images.
Data obtained at intervals of almost one second were connected to form the movie.
Blue, red and green regions indicate the monoclinic domains with different zigzag orientation.
Supplementary Figure 14 : Calculated phonon dispersion of LiVS 2 with a trigonal space group 3 ̅ 1. The most negative value of the frequency appears at the M point.