††thanks: Corresponding author.

# Zigzag chain order of LiVSe2 developing away from the vanadium trimer phase
transition boundary

K. Kojima  Department of Applied Physics, Nagoya University, Aichi 464-8603,
Japan  N. Katayama<EMAIL_ADDRESS>Department of
Applied Physics, Nagoya University, Aichi 464-8603, Japan  K. Sugimoto
Department of Physics, Keio University, Kanagawa 223-8522, Japan  N. Hirao
Diffraction and Scattering Division, Center for Synchrotron Radiation, Japan
Synchrotron Radiation Research Institute, Hyogo 679-5198, Japan  Y. Ohta
Department of Physics, Chiba University, Chiba 263-8522, Japan  H. Sawa
Department of Applied Physics, Nagoya University, Aichi 464-8603, Japan

###### Abstract

The phenomenon of self-assembly of constituent elements to form molecules at
low temperatures appears ubiquitously in transition metal compounds with
orbital degrees of freedom. Recent progress in local structure studies using
synchrotron radiation x-rays is shifting the interest in structural studies in
such molecule-forming systems from the low-temperature ordered phase to the
short-range order that appears like a precursor at high temperatures. In this
study, we discuss both experimentally and theoretically the relationship
between the trimer structure that appears in the layered LiV$X_{2}$ ($X$ = O,
S, Se) system with a two-dimensional triangular lattice of vanadium and the
zigzag chain-like local structure that appears near the phase transition
boundary where molecular formation occurs. The vanadium trimerization that
persistently appears in both low-temperature phases of LiVO2 and LiVS2
disappears in LiVSe2, and a regular triangular lattice is thought to be
realized in LiVSe2, but this study reveals that the zigzag chain local
distortion appears with a finite correlation length. This zigzag chain state
local distortions are similar to the motif of local distortions in the high-
temperature phase of LiVS2, indicating that the local distortions are
persistent away from the trimer phase transition boundary. On the other hand,
it is concluded that the zigzag chain order appearing in LiVSe2 is more stable
than that in LiVS2 in terms of the temperature variation of atomic
displacement and correlation length. The zigzag chain order is considered to
be competitive with the trimer order appearing in the LiV$X_{2}$ system. In
this paper, we discuss the similarities and differences between the parameters
that stabilize these electronic phases and the local distortions that appear
in other molecular formation systems.

††preprint: APS/123-QED

##  I Introduction

Since the discovery of the Verwey transition of magnetite in the 1930s [1],
many physical and structural studies have been devoted to transition metal
compounds that undergo molecular formation at low temperatures [2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. The molecular
formation phenomena in transition metal compounds often appear in pyrochlore
and triangular lattice compounds with high symmetry. High $d$-orbital
degeneracy and associated orbital degrees of freedom, the low dimensionality
of electronic structures such as hidden one-dimensionality produced by orbital
linkages, and competition between itinerancy and localization are considered
to be important factors in molecular formation phenomena [2, 3, 4, 5], and
together with drastic physical properties such as metal-insulator transition
and giant entropy change that appear with molecular formation [18, 22, 23,
24], they have attracted much attention from both basic and applied
perspectives. With recent progress in structural analysis techniques using
synchrotron radiation x-rays and neutrons, the interest in structural studies
of these molecular formation systems is shifting from elucidating the complex
molecular formation patterns that appear in the low-temperature phase to the
precursor local distortions that appear in the high-temperature phase. For
examples, in the spinel lattice system AlV2O4, vanadium heptamer (trimer +
tetramer), which appears at low temperatures, persists as short-range order at
high temperatures far beyond the phase transition [8], and in CuIr2S4, which
forms complex dimer patterns with charge ordering at low temperatures, short-
ranged tetragonal distortion appears in the high-temperature phase [25], as
revealed by a pair distribution function analysis (PDF) using synchrotron
radiation x-rays. The existence of such local distortions indicates that
orbital ordering is localized in the high-temperature phase, which was
believed to maintain high orbital degeneracy [25], and may affect our
understanding of the mechanism of molecular formation and the thermodynamics
associated with phase transitions.

Layered LiV$X_{2}$ ($X$ = O, S, Se) with a two-dimensional triangular lattice
provides a unique playground for studying the relationship between the
molecular formations that appear in the low-temperature phase and the local
distortions that appear in the high-temperature phase. The electronic phase
diagram characterizing the LiV$X_{2}$ system is shown in Fig. 1, which was
modified from Fig.1 in the Ref. [9] based on the zigzag chain order of LiVSe2,
which will be presented in this article later. LiVO2 is an insulator in the
whole temperature range and shows a nonmagnetic insulator transition at about
480 K. It has long been argued that a vanadium trimer state appears in the
low-temperature nonmagnetic insulating state of LiVO2, which has been fully
clarified by a recent PDF structural analysis based on the unique structural
model [26, 27]. Substitution of O(2$p$) for S(3$p$) or Se(4$p$) increases the
itinerancy. In LiVS2, the high-temperature phase becomes metallic while
maintaining the trimer in the low-temperature phase, and in LiVSe2, the trimer
state is completely suppressed, and the metallic state appears at all
temperatures.

Similar to AlV2O4 [8] and CuIr2S4 [25] mentioned above, LiVS2 also shows local
distortion on the triangular lattice of vanadium above the trimer transition
temperature of $T_{c}$ $\sim$ 314 K [28]. However, the characteristics of the
local structure of LiVS2 are very different from those of other materials that
undergo molecular formation. First, in the high-temperature phase of LiVS2, a
zigzag chain order appears preliminarily, which is clearly different from the
trimer motif in the low-temperature phase. Second, the correlation length of
the local distortion appearing in the conventional molecular formation system
is only a few Å, whereas the zigzag chain order appearing in the high-
temperature phase of LiVS2 has a long correlation length of several hundred Å.
Surprisingly, the correlation length increases further just above the trimer
transition and superlattice peaks appear, which originate from the zigzag
chain ordering. Third, this zigzag chain distortion appears to exhibit
spatially and temporally fluctuating dynamics when observed by time-resolved
Scanning Transmission Electron Microscope (STEM) measurements.

It has been argued that this anomalous local distortion is related to the
appearance of pseudogaps, a phenomenon of anomalous reduction in magnetic
susceptibility that appears in the metallic phase near the trimer phase
transition boundary in terms of physical properties [9, 28]. If the zigzag
chain order originates from the V3+($d^{2}$) electronic state and implies bond
formation between neighboring vanadium ions, it is understood that the
evolution of correlation length implies a gradual increase in the number of
$d$ electrons involved in spin-singlet formation, leading to a pseudogap-like
decrease in magnetic susceptibility. Based on this idea, LiVSe2 could be an
attractive research target to investigate the development of zigzag chain
order in LiV$X_{2}$ ($X$ = O, S, Se) phase space. This is because LiVSe2
exhibits metallic conduction over the entire temperature range, and this
metallic phase is connected to the metallic phase of LiVS2 in the phase space
where the zigzag chains appear [9]. More interestingly, according to the
temperature dependence of the magnetic susceptibility of LiVSe2 reported
previously [29], the Pauli paramagnetic component of the magnetic
susceptibility obtained by subtracting the Curie tail has a positive slope
with respect to temperature, seems to indicate that a pseudogap similar to
that of LiVS2 appears persistently in LiVSe2. It should be noted that a
similar temperature dependence of magnetic susceptibility has been observed
for our LiVSe2 samples, the details of which are summarized in Supplemental
Material [30]. By clarifying the appearance of zigzag chains in LiVSe2,
examining their stability with respect to temperature and pressure, and
comparing them with the characteristics of zigzag chains in LiVS2, it is
expected that the relationship between the trimer phase and zigzag chain order
can be studied in detail.

Figure 1:  Phase diagram of the LiV$X_{2}$ system. Although the classification
of each phase has been shown in previous studies [9, 28], our study reveals
that the zigzag chain state shown in green appears more stable in LiVSe2,
which is farther from the trimer phase boundary compared to LiVS2. The
different colored dotted lines on the inset figures indicate different V-V
distances.

In this paper, we discuss the relationship between the zigzag chain ordering
and the trimer phase in the LiV$X_{2}$ ($X$ = O, S, Se) system, focusing on
the results of structural studies obtained utilizing synchrotron radiation
x-rays and theoretical calculations of LiVSe2. Synchrotron x-ray diffraction
experiments have revealed that the zigzag chain order appears in LiVSe2 with a
finite correlation length, as in the high-temperature phase of LiVS2.
Strangely, even though LiVSe2 is farther from the trimer transition boundary
in phase space, the zigzag chain order has a larger vanadium displacement than
that of LiVS2 and persists at higher temperatures with longer correlation
lengths. This indicates that the zigzag chain local distortion that appears in
the LiV$X_{2}$ system is not merely a precursor to the trimerization that
appears at low temperatures. Based on theoretical calculations and pressure
effects, we discuss the factors that stabilize the trimer and zigzag chains,
and argue that the LiV$X_{2}$ system is the stage for studying a unique local
distortion phenomenon that is clearly different from conventional molecular
formation systems.

##  II Methods

###  II.1 Sample synthesis

LiVSe2 was synthesized by a three-step synthesis method. First, the raw
materials, V (99.5%) and Se (99.999%), were mixed so that V:Se = 1:1.01 to
avoid V self-intercalation, sealed in a vacuum quartz tube, and calcined at
700 $\tccentigrade$ for 3 days to obtain 1$T$-VSe2. The resulting powder
sample was then immersed in a large excess of 0.2 $M$ $n$-BuLi/hexane solution
for 2 days in a glove box under an Ar atmosphere. This yields Li2VSe2. To
remove excess Li, the sample was immersed in I2-acetonitrile; by adjusting the
weight of I2, the deintercalation of Li can be quantitatively controlled. The
resulting samples were annealed at 150 $\tccentigrade$ for 10 hours to restore
crystallinity and obtain powder samples of LiVSe2. For LiVO2 and LiVS2, we
used the same batch of samples as presented in the previous paper [26].

###  II.2 Experiments

Inductively coupled plasma (ICP) measurements were performed to calculate the
Li content of the synthesized samples. Considering the presence of impurity
Li2Se (3.4%) revealed by Rietveld analysis, ICP measurement estimated the
LiVSe2 used in this study to be Li/V = 1.01. The occupancy of the Se site was
0.983(3) as a result of our Rietveld analysis. Differential scanning
calorimetry (DSC) was conducted using a DSC 204 F1 Phoenix (Netzsch).

The ambient pressure x-ray diffraction data were measured at BL44B2 and BL02B2
at SPring-8. 30 keV x-rays and the detectors commonly used at each beamline
were used. The data obtained at BL44B2 were corrected, combined, and converted
to one-dimensional data [31, 32]. The x-ray diffraction experiments under high
pressure were performed at BL10XU at SPring-8 [33]. A diamond anvil cell (DAC)
was used for high-pressure generation. The sample was loaded into the DAC,
together with some rubies as pressure markers [34]. Helium was used as a
pressure-transmitting medium. 30 keV x-ray energy was used, and diffraction
data were acquired with an imaging plate detector. The obtained diffraction
data were indexed using Conograph [35], and the analysis was performed using
Rietan-FP [36]. The obtained structures were drawn using VESTA [37].

X-ray absorption fine structure (XAFS) measurements were performed on the
vanadium $K$-edge ($\sim$ 5.47 keV) at BL5S2 and BL11S2 at Aichi Synchrotron
Radiation Center. XAFS spectra were collected by the Quick-XAFS method using a
vanadium foil as a reference. The samples were mixed with BN and pelletized.
Temperatures below 300 K were controlled by a cryostat, while temperatures
above 300 K were controlled by a heater. The obtained data were transformed
using Athena included in Demeter [38]. For the Fourier transform, the range up
to 12 $\mathrm{\AA}^{-1}$ was used for the almost data, and for the 30 K data
of LiVO2, the range up to 13 $\mathrm{\AA}^{-1}$ was used so that the
interatomic distances originating from the trimer would be apparent.

###  II.3 Theoretical calculations

The partial densities of states (PDOS) and the Fermi surfaces in LiVSe2 and
LiVS2 were calculated by employing the WIEN2k code [39, 40] based on the full-
potential linearized augmented plane-wave method. These results are obtained
using the generalized gradient approximation (GGA) for electron correlations,
where the exchange-correlation potential of reference [41] is used. To improve
the description of electron correlations in V 3$d$ orbitals, we use the
GGA+$U$ method with the around-the-mean-field double-counting scheme [42],
which is believed to be suitable for metallic systems. In the following, we
will present the results obtained at $U$ = 4 eV. PDOS for the triangular
lattice of LiVS2 and LiVSe2 were calculated using lattice parameters obtained
by analyzing LiVS2 data at 400 K and LiVSe2 at 300 K assuming trigonal space
group $P\bar{3}m1$. We also calculated the PDOS for the zigzag chain structure
of both materials using the lattice parameters obtained by analyzing the data
of LiVS2 at 320 K and LiVSe2 at 100 K assuming the original space group, the
monoclinic space group $Pm$. It should be noted that monoclinic distortions of
these materials were small at high temperatures and could be indexed by the
trigonal space groups, as discussed below. In the self-consistent
calculations, we use $12\times 12\times 6$ $\bm{k}$-points in the Brillouin
zone. The muffin-tin radii ($R_{\mathrm{MT}}$) for the $P\bar{3}m1$ structure
are taken as 2.23 (Li), 2.45 (V), and 2.11 (S) Bohr for LiVS2 and 2.14 (Li),
2.4 (V), and 2.4 (Se) Bohr for LiVSe2. Similarly, the $R_{\mathrm{MT}}$ for
the $Pm$ structure are taken as 2.16 (Li), 2.29 (V), and 2.07 (S) Bohr for
LiVS2 and 2.11 (Li), 2.3 (V), and 2.3 (Se) Bohr for LiVSe2. The plane-wave
cutoff is set to $K_{\mathrm{max}}=7.5/R_{\mathrm{MT}}$.

##  III Results and discussion

###  III.1 X-ray diffraction experiments at ambient pressure

Figure 2:  (a,b) Rietveld analysis at 100 K for assuming (a) a triangular
lattice and (b) a zigzag chain structure. The insets show the area of the
green box. The green ticks are the peak positions of the structure used for
fitting, and the gray ticks are for impurity Li2Se.

Figure 2(a) shows the results of Rietveld analysis at 100 K performed assuming
a triangular lattice structure of LiVSe2 with space group $P\bar{3}m1$ and the
presence of Li2Se impurities. Although the simulation pattern seems to
reproduce the diffraction data well, some peaks could not be indexed based on
the structure model with a space group $P\bar{3}m1$. Assuming the structure of
the monoclinic space group $Pm$ with vanadium forming zigzag chains, all
peaks, including these unknowns, were fitted well, as shown in Fig. 2(b). The
obtained lattice parameters are $a_{\mathrm{m}}$ = 6.2407(6) $\mathrm{\AA}$,
$b_{\mathrm{m}}$ = 3.5632(3) $\mathrm{\AA}$, $c_{\mathrm{m}}$ = 6.3425(2)
$\mathrm{\AA}$, $\beta$ = 89.884(14)∘, $\alpha$ = $\gamma$ = 90∘. As can be
seen from the fact that $a_{\mathrm{m}}/\sqrt{3}b_{\mathrm{m}}$ = 1.011 is
close to 1 and $\beta$ is close to 90∘, the unit cell shape does not clearly
show monoclinic distortion, which is the reason why diffraction pattern is
fitted fairly well using the trigonal space group $P\bar{3}m1$. However, as
indicated by the appearance of distinct superlattice peaks, the formation of
the zigzag chains splits the V-V distance into three different types, as shown
in the inset of Fig. 1.

Figure 3:  (a) Temperature dependence of the x-ray diffraction data of LiVSe2
obtained at ambient pressure. The indices of each peak listed in the figure
assume a triangular lattice structure. (b) Intensity ratios of the peaks
$\bar{\frac{1}{2}}$12 + $\frac{1}{2}$10 (blue) and $\frac{1}{2}$03 +
$\frac{1}{2}$0$\bar{3}$ (red) in LiVSe2 and LiVS2 (Ref. [28]) to the 001
fundamental peak. The values due to LiVS2 are so small that they are enlarged
and shown on the right. (c) Temperature dependence of the V-V distance
obtained by Rietveld analysis. The colors plotted for each interatomic
distance correspond to the dotted lines shown in the structure diagram to the
right of the graph.

As shown in Fig. 3(a), the intensity of the superlattice peaks that appear due
to monoclinic distortion gradually decrease with increasing temperature and
are no longer observed as a distinct peak above 500 K. For quantitative
evaluation, the intensities of the peaks at $\bar{\frac{1}{2}}$12 +
$\frac{1}{2}$10 and $\frac{1}{2}$03 + $\frac{1}{2}$0$\bar{3}$ normalized by
the 001 peak are shown in Fig. 3(b). The intensities of the superlattice peaks
decay linearly with increasing temperature, reaching zero at around 500 K.
Corresponding to the decay of these intensities, the splits of in-plane V-V
interatomic distance shown in Fig. 3(c) gradually decrease with increasing
temperature.

The structural features of the zigzag chain state of LiVSe2 shown above are
similar to those of the high-temperature metallic phase of LiVS2 [28]. In
LiVS2, even just above the metal-insulator transition temperature of 314 K,
where the monoclinic distortion related to the zigzag chain is largest,
$\beta$ is 89.956(3)∘ and $a_{\mathrm{m}}/\sqrt{3}b_{\mathrm{m}}$ retains to
1.001, which is the indicator of the magnitude of the monoclinic distortion.
Thus, the monoclinic distortion feature is hardly apparent in the unit cell
shape. Nevertheless, a large vanadium displacement of $\sim$0.13
$\mathrm{\AA}$ appears with zigzag chain formation.

The reason for these strange structural features appearing in LiVS2 is related
to the fact that the zigzag chain order is not long-range enough, and the
correlation length remains finite. This is evident from the experimental
results in Fig. 1(a) in the reference [28], where the vanadium displacement
estimated by the average structure is always smaller than the local vanadium
displacement estimated by the pair distribution function analysis (PDF
analysis). If the correlation length of the zigzag chain is long enough, the
magnitude of the vanadium displacement obtained from the average structural
analysis should be equal to that obtained from the local structural analysis.
These features are also evident in the shape of the unit cell as estimated by
the average structure. If such a large vanadium displacement occurred, the
unit cell would naturally be more distorted from trigonal. However, if the
monoclinic distortion associated with vanadium displacement is only localized
and the monoclinic domains are oriented in various directions with short
correlation lengths, the average structure analysis will appear as if a
trigonal lattice is realized. In LiVS2, such monoclinic distortions are
developed with “mesoscopic” correlation lengths of a few hundred Å or more, so
to speak, which is why they are observed as such strange structural features.

###  III.2 XAFS experiments on LiV$X_{2}$ ($X$ = O, S, Se) series

It is interesting that structural features similar to LiVS2 were observed in
LiVSe2. This seems to suggest that the same features are realized in LiVSe2 as
in LiVS2, where the correlation length of the zigzag chain remains finite and
shortens with increasing temperature. To confirm this, we performed the
vanadium $K$-edge XAFS measurements of three LiV$X_{2}$ compounds ($X$ = O, S,
and Se) at various temperatures. Fig. 4(a) shows the temperature dependencies
of the radial distribution function (RDF) for three samples. This is a
function of the probability of an atom at a distance $r$ from any given
vanadium in the crystal structure, obtained by the Fourier transform of the
XAFS spectrum. In the LiVS2 data at 30 K, a peak corresponding to the V-S
distance appears at $\sim$2 $\mathrm{\AA}$, and two peaks corresponding to the
V-V distances associated with vanadium trimerization appear from 2.5 to 3
$\mathrm{\AA}$, as shown in Fig. 1. It should be noted that the presence of
two separate peaks has already been discussed in reference [9]. Strangely
enough, once the temperature is raised to the high-temperature phase, the
peaks corresponding to the V-V distance disappear completely, and only the
peaks related to the V-S distance appear.

The absence of a peak corresponding to the V-V distance is consistent with the
previous paper [9], but the cause is not mentioned there. This may be since
the correlation length of the zigzag chains of vanadium in the high-
temperature phase LiVS2 is much smaller than the beam diameter of 500 $\mu$m
and shows fluctuating dynamics in time and space. These dynamics occur on the
order of seconds, which is well faster than the measurement time of 60 s for
XAFS spectra. Because of the relationship between the features exhibited by
zigzag chains and the experimental conditions, the addition of spectra
obtained from multiple zigzag chain structures with different orientations is
observed in XAFS measurements. Then, due to the phase cancellation of the XAFS
spectra, the peaks corresponding to the V-V distance are lost in the RDF. It
should be noted that exactly the same RDF results are obtained at 325 K, where
the displacement of vanadium from the zigzag chains is clearly observed in the
average structure, and at 400 K, where no displacement is observed. This
suggests that the local structure remains unchanged as the temperature
increases and that the vanadium lattice similarly fluctuates at 325 and 400 K.

Figure 4:  (a) The RDFs by XAFS measurements on LiV$X_{2}$. (b) Results of DSC
measurements in LiVSe2 and reference LiVS2; results of DSC measurements in
LiVS2 are equal to those reported in previous studies [28].

Similarly, in the spectrum of LiVSe2, a peak corresponding to the V-Se
distance is observed near $\sim$2.2 $\mathrm{\AA}$, but no peak corresponding
to the V-V distance is observed at all. If the zigzag chain structure in
LiVSe2 forms a long-range order without fluctuating, three types of split V-V
distances should be observed in RDF. Therefore, the present result indicates
that the zigzag chain interactions are not sufficiently long-range ordered and
disordered as in LiVS2. In addition, interestingly, no clear change in the RDF
appears well beyond around 500 K, where the superlattice peaks disappear in
x-ray diffraction data. This clearly suggests that the zigzag chain order
persists to high temperatures while becoming short-range ordered in LiVSe2 as
well.

It should be noted that the correlation length of the zigzag chain changes
continuously with temperature, and therefore no clear signature is observed in
the physical property measurements. As shown in Fig. 4(b), DSC measurements of
LiVSe2 performed in the range of 300 - 650 K show no anomaly around 500 K.
This is also the case in LiVS2. In the DSC data of LiVS2, the superlattice
peaks disappear without any sign around 350 K, indicating that the cause of
the disappearance of the superlattice peaks is similar in both cases.

The above experimental results might seem to indicate that the zigzag chains
that appear as precursors to trimerization in LiVS2 also persist in LiVSe2,
but this is not such a simple case. If the zigzag chain order is best
developed near the trimer phase transition boundary, then the zigzag chain
order of LiVS2, which shows trimerization at low temperatures, should be more
stable than that of LiVSe2. However, as shown in Figs. 3(b) and (c), the
zigzag chain order in LiVSe2 is stable at higher temperatures with larger
vanadium displacements. It should be pointed out that in LiVO2, which has the
highest trimer transition temperature in the LiV$X_{2}$ series, no zigzag
chain order is reported to appear at high temperatures [26]. In fact, as shown
in Fig. 4(a), XAFS measurement of LiVO2 clearly shows a single V-V distance
peak in the high-temperature phase as well as two V-V distances originating
from the trimerization that appear in the low-temperature phase. The
appearance of a peak corresponding to the V-V distance in the high-temperature
phase is also reported in Ref. [43]. This indicates that the vanadium lattice
disorder that appears in LiVS2 and LiVSe2 does not occur in the high-
temperature phase of LiVO2. This indicates that this anomalous zigzag chain
state appears universally and stably over a range of the metallic phase of
LiVS2 and LiVSe2, not only in the vicinity of the trimer phase, as shown in
Fig. 1.

###  III.3 Partial DOS calculations of LiVSe2

The above structural studies show that zigzag chain ordering is realized in
both metallic phases of LiVS2 and LiVSe2, persistently surviving with the
correlation length becoming shorter when the temperature increases. A unique
feature of this system is that the zigzag chains are more stabilized at
LiVSe2, which is farther from the metal-insulator transition boundary than
LiVS2 in the phase space shown in Fig. 1. Partial DOS calculations were
performed to determine the cause of these characteristics. Figure 5(a) is the
DOS computed on the basis of the space group $P\bar{3}m1$ realized as an
average structure at high temperatures in LiVS2 and LiVSe2. The calculation
result for LiVS2 is essentially identical to that shown in [28]. LiVS2 and
LiVSe2 have similar DOS shapes with a large population due to the 3$d$
orbitals of vanadium near the Fermi surface. However, due to the difference in
lattice parameters associated with the difference between S(3$p$) and
Se(4$p$), the energy width of the $d$ orbitals is slightly narrower in LiVSe2.
This will be important in later discussions.

On the other hand, Fig. 5(b), calculated for the space group $Pm$ structure
with zigzag chain structure, shows a clear decrease in the density of states
near the Fermi surface for both LiVS2 and LiVSe2. This happens to be due to
the nesting of the Fermi surfaces that occurs as a result of zigzag chain
formation. In the structure of the space group $P\bar{3}m1$ of LiVS2, it has
been discussed that there is an instability of the Fermi surface defined by
the nesting vector $Q$ = $a^{*}$/2 as shown in Fig. 5(c), which causes zigzag
chain formation corresponding to vanadium displacements. A similar instability
of the Fermi surface is expected to exist in LiVSe2, which may lead to the
zigzag chain formation in LiVSe2. However, the Fermi surfaces have not
completely disappeared in both cases, which is consistent with the high-
temperature phase of LiVS2 and LiVSe2 being essentially metallic.

Figure 5:  (a) Partial-DOSs of LiVS2 (upper) and LiVSe2 (lower) assuming a
triangular lattice structure. (b) Partial-DOSs of LiVS2 (upper) and LiVSe2
(lower) assuming a zigzag chain structure. (c) Shapes of Fermi surfaces for
LiVS2 (upper) and LiVSe2 (lower) with a triangular lattice structure. The
figures of LiVS2 are equal to those shown in the previous study [28]. (d)
Relationship between $d_{xy}$, $d_{yz}$, and $d_{zx}$ orbitals. Electrons
occupy 1/3 of each orbital. (e) Zigzag chain formation due to the change of
filling by charge transfer.

The reason why the zigzag chain appears more stable in LiVSe2, even though the
partial DOS of LiVS2 and LiVS2 are very similar, may be due to the difference
in the energy width of the $d$ orbital. In the trigonal structure, the
$d_{xy}$, $d_{yz}$, and $d_{zx}$ orbitals that consist of $t_{2g}$ orbitals
are triply degenerate, and the electron filling of these three orbitals are
equal to 1/3, as shown in Fig. 5(d). If the triangular lattice is distorted to
form a zigzag chain, in principle the electron filling must change from
($d_{xy}$,$d_{yz}$,$d_{zx}$)=(1/3,1/3,1/3) to a filling of
($d_{xy}$,$d_{yz}$,$d_{zx}$)=(0,1/2,1/2), as shown in Fig. 5(e). This has been
pointed out by Rovira and Whangbo [5], who argued that narrower $t_{2g}$
orbitals require less energy for electron transfer, thus stabilizing the
zigzag chain state. LiVS2 and LiVSe2 actually maintain metallic conduction in
the zigzag chain state. Therefore, ($d_{xy}$,$d_{yz}$,$d_{zx}$)=(0,1/2,1/2) is
not strictly realized and some electrons are considered to remain in the
$d_{xy}$ orbitals. However, the difference in bandwidth as pointed out by
Rovira and Whangbo is found in our theoretical calculations, as shown in Fig.
5(a), which may be the reason why more stable zigzag chains appear in LiVSe2.

###  III.4 X-ray diffraction under high pressure

Figure 6:  (a) Pressure dependence of diffraction data. The figure shows
diffraction data from ambient pressure to 40 GPa in increments of about 5 GPa.
An enlarged view of the low-angle data is shown below. (b) Pressure
dependencies of the lattice parameters and $c/a$ at 8 K. $c/a$ for VSe2 is
taken from R. Sereika $et~{}al.$ [44] as a reference. (c) Pressure dependence
of the estimated in-plane V-V distance. (d) Temperature dependence of averaged
V-S distance in LiVS2.

Since the zigzag chain formation is associated with the instability of the
Fermi surface, it is expected that applying pressure causes poor nesting of
the Fermi surface. Instead, trimerization is stabilized by the effect of
closer V-V distance and stronger $d$-$d$ hybridization. Based on these
expectations, we performed diffraction experiments under pressure. Fig. 6(a)
shows the diffraction data at several pressure points at 8 K. When pressure is
applied, the diffraction peak clearly shifts to the high-angle side,
indicating that the lattice parameter is decreased, as summarized in the Le
Bail analysis results in Fig. 6(b). The pressure dependence of the diffraction
data in Fig. 6(a) shows that the intensity of the superlattice peaks from the
zigzag chain structure gradually increases with increasing pressure,
suggesting that, contrary to expectations, the zigzag chain state stabilizes
under pressure. The Rietveld analysis is necessary to estimate the amount of
atomic displacement of V that causes zigzag chains, but it is difficult to
perform accurate analyses because of the reduced crystallinity and higher
background under pressure. Therefore, we used the following method to estimate
the atomic positions in the crystal. First, Rietveld analysis was performed on
the data at 20.8(1) GPa, where the peaks are relatively sharp, and the
superlattice intensity is reasonable. Using the structural parameters obtained
from this analysis and the ratio of fundamental and superlattice intensity, we
estimated the structural parameters for each data. The Supplemental Material
[30] describes the method’s details. As shown in Fig. 6(c), the splitting of
the V-V interatomic distance at 8 K increases continuously under pressure,
reaching as much as 0.33 $\mathrm{\AA}$ (10.6%) at 40.0(1) GPa. Furthermore,
as shown in Fig. 3(c), while the V-V interatomic distance shows a strong
temperature dependence at ambient pressure, there is no difference in the
magnitude of displacement between 8 K and 300 K under pressure, indicating
that the zigzag chains are rapidly stabilized under pressure.

There are two possible reasons for the unexpected stabilization of the zigzag
chains. First, the Li ions between the VSe2 layers prevented the
dimensionality from decreasing under pressure. This is evident from the
pressure dependence of $c/a$, a measure of dimensionality, shown in the lower
part of Fig. 6(b). Generally, the $c/a$ value decreases rapidly with applied
pressure in layered compounds because the van der Waals gap between layers is
easily compressed. In VSe2, which is included for reference, the value of
$c/a$ decreases with pressure [44], but in LiVSe2, contrary to expectations,
it does not change. This behavior may be due to the insertion of Li ions
between the VSe2 layers, which prevents the van der Waals gap from
compressing. The presence of Li in LiVSe2 not only sets the stage for the
formation of zigzag chains by changing vanadium to the $d^{2}$ electronic
state, but also prevents the zigzag chains from becoming unstable under
pressure by losing the two-dimensionality of the electronic structure, which
is important for Fermi surface nesting. Second, the V-S distance shrinks under
pressure, and $d$-$p$ hybridization is strengthened. As shown in Fig. 6(d), in
LiVS2 where vanadium trimerization appears at low temperatures, the average
V-S distance is clearly increased by trimerization. This would indicate that
trimerization appears from strong $d$-$d$ hybridization between ionic
V3+($d^{2}$) and therefore does not favor strong $d$-$p$ hybridization. Under
high pressure in LiVSe2, both $d$-$d$ and $d$-$p$ hybridization are enhanced.
Although these two effects should be competing, the effect of reduced
V-ionicity due to increased $d$-$p$ hybridization is more dominant in LiVSe2,
resulting in the trimer state not appearing under high pressure in LiVSe2.
Instead, the zigzag chain state stabilizes under high pressure.

###  III.5 Discussion

From the above experiments, it seems likely that the zigzag chain states
appearing in the LiV$X_{2}$ system are not simply trimer precursors, but are
characterized as competing electronic phases governed by various parameters
such as Fermi surface nesting, band width, $d$-$p$ and $d$-$d$ hybridization.
On the other hand, however, the feature that the correlation length of the
zigzag chain state remains finite up to high temperatures indicates that the
zigzag chain state is very different from the usual electronic phase, rather
similar to the local distortion features that appear in the high-temperature
phase of many systems where molecular formation occurs at low temperatures
[45, 46, 47, 13, 8, 48, 49]. There seem to be several cases of local
distortion phenomena that appear at high temperatures: First, the case in
which molecular formation patterns that appear at low temperatures appear as
short-range order. This is the most common case and has been discussed in
Li2RuO3 [46], AlV2O4 [8], LiRh2O4 [13], RuP [47], etc. The recently reported
dimer fluctuation of 1$T$-$M$Te2 ($M$ = V, Nb, Ta) may also be a variant of
this case [48]. Second, the case in which distortion patterns different from
those of the low-temperature phase appear as short-range order. A typical
example is CuIr2S4 [25], where it is argued that the dimer of the low-
temperature phase does not appear as a short-range order, but instead the
tetragonal distortion appears as a short-range order. Third, the case of
LiV$X_{2}$ ($X$ = S, Se), where a completely different pattern from the low-
temperature phase appears with a finite correlation length. Among these, the
LiV$X_{2}$ system is also unusual in that it forms an order with a long
correlation length on the mesoscopic scale, and it undoubtedly provides a
unique playground that sets it apart from other material systems that exhibit
molecular formation. Through further exploration of materials forming short-
range order, we expect that the physics of these molecule-forming systems will
develop into an important research field related to various fields, including
quantum liquid crystals and nonequilibrium physics.

##  IV Summary

In summary, we have performed structural studies and theoretical calculations
of layered LiV$X_{2}$ and found that the zigzag chain state that appears in
the high-temperature phase of LiVS2 is more stabilized in LiVSe2. This
indicates that the zigzag chain phase appears more stable in the phase space
of LiV$X_{2}$ ($X$ = O, S, Se) in regions away from the trimer phase boundary.
The zigzag chain phases found in this system are significantly different from
the local distortion features that appear in conventional molecular formation
systems, clearly indicating that molecular formation systems with a long
history, as typified by the Verwey transition of magnetite, are still unique
playgrounds with a rich variety of fascinating physics.

###### Acknowledgements.

We are very grateful to R. Isoda and Dr. R. Ishii for their experimental
support. The work leading to these results has received funding from the Grant
in Aid for Scientific Research (No. JP17K17793, No. JP20H02604, No.
JP20H01849, No. JP21K18599, No. JP21J21236, No. JP22KJ1521, and No.
JP23H04104) and Research Foundation for the Electrotechnology of Chubu. This
work was carried out under the Visiting Researcher’s Program of the Institute
for Solid State Physics, the University of Tokyo, and the Collaborative
Research Projects of Laboratory for Materials and Structures, Institute of
Innovative Research, Tokyo Institute of Technology. Powder x-ray diffraction
experiments were conducted at the BL02B2, BL10XU, and BL44B2 of SPring-8,
Hyogo, Japan (Proposals No. 2020A1063, No. 2021A1111, No. 2021A1117, No.
2021B1136 and No. 2022A1167), and at the BL5S2 of Aichi Synchrotron Radiation
Center, Aichi Science and Technology Foundation, Aichi, Japan (Proposals No.
202105170, No. 202106066, No. 2021L1002, and No. 2020L6002). XAFS experiments
were conducted at the BL5S1 and BL11S2 of Aichi Synchrotron Radiation Center
(Proposals No. 201901017, 201902049, 202201031, and 202302018).

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