# Raman study of the structural transition in LiVO2

Yuri S. Ponosov  Institute of Metal Physics, Ural Branch of the Russian
Academy of Sciences, Ekaterinburg 620137, Russia  Evgenia V. Komleva
Institute of Metal Physics, Ural Branch of the Russian Academy of Sciences,
Ekaterinburg 620137, Russia  Department of theoretical physics and applied
mathematics, Ural Federal University, Ekaterinburg 620002, Russia  Elizaveta
A. Pankrushina  Laboratory of Arctic Mineralogy and Material Sciences, Kola
Science Centre, Russian Academy of Sciences, Apatity 184209, Russia  Haohang
Xu  School of Physics, Harbin Institute of Technology, Harbin 150001, China
Yu Sui  School of Physics, Harbin Institute of Technology, Harbin 150001,
China  Laboratory for Space Environment and Physical Sciences, Harbin
Institute of Technology, Harbin 150001, China  Sergey V. Streltsov  Institute
of Metal Physics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
620137, Russia  Department of theoretical physics and applied mathematics,
Ural Federal University, Ekaterinburg 620002, Russia

###### Abstract

The results of polarization-dependent Raman spectroscopy of single-crystalline
LiVO2 exhibiting transition to a diamagnetic state below $T_{c}\sim$500K are
reported. Our measurements clearly detect additional peaks in the low-
temperature phase, which disappear nearly completely when heated above
$T_{c}$. Proposed $\sqrt{3}$a × $\sqrt{3}$a lattice reconstruction explains
these new Raman peaks by the Brillouin zone folding. The experiment, on the
one hand, confirms that the symmetry of the non-magnetic phase is not lower
than trigonal, but, on another hand, our thermal cycling study suggests
possible stacking faults. This agrees with results of the density functional
theory calculations, which show that the energy difference between different
types of stacking does not exceed 1 K per formula unit.

##  I Introduction

LiVO2 belongs to class of materials with a strong interplay of orbital,
lattice, and spin degrees of freedom [1]. It crystallizes in a rhombohedral
structure with $R\bar{3}m$ space group above 500 K [2]. Structurally, it
consists of alternating Li/V and O layers and within each layer the V ions
form a lattice of triangles, which are equilateral in the high-temperature
phase. These ions are octahedrally coordinated and therefore V 3$d$ states are
split onto doubly degenerate $e_{g}$ and triply (nearly) degenerate $t_{2g}$
levels, due to a ligand field of approximate $O_{h}$ symmetry.

Almost seventy years ago this material was found to undergo a first-order
structural transition at $T_{c}\sim 500$ K [3]. This transition is accompanied
by drastic changes of magnetic properties. While magnetic susceptibility
follows Curie-Weiss law at high temperatures, the low-temperature phase turned
out to be non-magnetic [2]. Similar transitions have been observed in
LiVS2[4], Na2Ti3Cl8 [5; 6; 7], CsW2O6 [8; 9; 10; 11] and some other materials.
Several possible mechanisms have been proposed to explain the phase transition
observed in LiVO2. Goodenough argued that the in-plane V-V separation in this
material is near a critical distance $R_{c}$ that corresponds to the
situation, when the bandwidth $W$ is approximately equal to the on-site
Hubbard repulsion $U$. According to Goodenough’s phenomenological theory [12]
there develops an electronic instability that is manifested by the formation
of V3 trimers below $T_{c}$ [12]. An alternative scenario for the phase
transition in LiVO2 involves an orbital ordering of the localized V $3d^{2}$
electrons [13; 14]. The orbital degrees of freedom suppress frustration on
triangular lattice formed by V ions. Moreover, one might also think of the
orbitally driven Peierls transition [1]. All proposed mechanisms assume
formation of a spin-singlet state in the low-temperature phase due to
formation of vanadium trimers.

Indeed, superlattice reflections tentatively related to the trimerization were
observed by X-ray diffraction of powder samples [15; 16; 17; 18; 19]. The
$P31m$ space group was considered as a reasonable choice for the low-
temperature phase [15]. Superstructural spots appearing at (1/3, 1/3, 0) in
the diffraction patterns of LiVO2 single crystals were observed in [20; 21;
22; 23]. Structural changes at 500K transition were later seen by transmission
electron microscopy (TEM) experiments performed on LiVO2 single crystals [24].
Based on these experiments various space groups were proposed for the
trimerized phase $P31m$[15], $P3_{1}12$, $P3_{2}12$ [21; 22; 23], $P6_{2}22$,
$P6_{4}22$ [20]. In addition, it was noted that the configuration of trimers
in different layers can be different, even leading to their completely
disordered state along the $c$ axis, and in [21; 22; 23] a structure
(including 6 layers of vanadium) with doubling of the $c$ axis was suggested.

The possible tripling of the unit cell was predicted by theoretical
calculations, where the high-temperature phase was found to be unstable [25].
Frequencies of two acoustic modes of the phonon spectrum at $\bf
q=$(1/3,-1/3,0) with respect to the reciprocal lattice vectors of the
primitive unit cell turned out to be negative. This led Subedi to suggestion
that the primitive cell of the low-temperature phase with three formula units
(12 atoms) has a monoclinic $Cm$ structure with the V ions forming trimers
shown in Fig.1(b). Another version of the low-temperature crystal structure
was proposed in [16], and then refined taking into account the pair
distribution function (PDF) analysis in [19]. In a very recent PDF study by
Kojima et al. it was shown that apparent difficulties in crystal structure
determination in LiVO2 are related with a random stacking of V trimers along
$c$ direction [17; 26]. Thus, even after more than 70 years of investigations
the crystal structure of this material remains unsolved and, moreover, there
are indications that stacking faults are always present and one should rather
discuss presence of a short-range trimer order in the stacking direction [26].

This makes studying of LiVO2 local structure especially important. While Raman
spectroscopy is known to be extremely sensitive to local changes of the
crystal structure, surprisingly no detailed analysis of Raman data for LiVO2
have been performed so far. In this work, we measured symmetry-dependent Raman
spectra of LiVO2 three single crystals and study their temperature behavior
across the phase transition. Experimental results are supplemented by the
density functional theory (DFT) calculations of the phonon spectra and
possibility of the stacking faults. The results showed the surprising
sensitivity of Raman spectroscopy to structural changes in the sample, thereby
confirming the formation of superstructures due to trimerization of V ions.

Figure 1:  (a) Triangular lattice formed by V ions (shown in red) in the $ab$
plane in the high-temperature ($R\bar{3}m$) phase of LiVO2 and (b) the
resulting trimerized structure in the low-temperature phase of LiVO2 with V
trimers (corresponding short V-V bonds are shown in purple thick lines). Thin
red lines demonstrate longer V-V bonds.

##  II Experimental and calculation details

Small single crystals (of size 2$\times$2$\times$0.5 mm3) of LiVO2 were grown
by using the flux method as reported before [24]. The conventional solid-
state-reaction method was applied to prepare LiVO2 powder with pure phase, in
which appropriate proportions of Li2CO3 and V2O3 were thoroughly mixed and
then calcined at 625∘C and 750∘C under flowing Ar: H2 = 19: 1 with
intermediate grinding. Then, the mixture of LiVO2 polycrystalline, Li2O, and
LiBO2 was placed in a platinum crucible in the glove box and sealed in a
quartz tube. After heating at 1100∘C for 12 h, the tube was cooled to 700∘C by
oscillating process, then turned off the power. The chemical composition of
the sample was checked by the Inductively coupled plasma (ICP) mass
spectroscopy (NexION 350X), giving the ratio of Li: V = 1.01. The phase purity
was identified by the X-ray diffraction (XRD, Aeris, CuK$\alpha$1 radiation)
and the X-ray Laue back diffraction was used to confirm the quality of the
crystal, see Fig. 2.

Figure 2:  (a) X-ray diffraction (XRD) pattern and (b) Laue picture along the
(001) direction of grown LiVO2 single crystal.

We performed Raman measurements in the temperature range of 80 to 600 K in
backscattering geometry. The measurements were performed with RM1000 Renishaw
microspectrometer equipped with a 532 nm solid-state laser and 633 helium–neon
laser and with a thermocooled CCD camera. Respective Linkam stage was used for
temperature variation. Very low power (up to 1 mW) was used to avoid local
heating of the sample. A pair of notch filters with a cut-off at 40 cm-1 were
used to reject light from the 633 nm laser line. To reach as close to the zero
frequency as possible, we used a set of three volume Bragg gratings (VBG) at
532 nm excitation to analyze the scattered light. The resolution of our Raman
spectrometer was estimated to be 2–3 cm-1.

All the density functional theory (DFT) calculations were performed in Vienna
ab initio simulation package (VASP) [27]. We utilized generalized gradient
approximation (GGA) as proposed by Perdew, Burke, and Ernzerhof (PBE) [28] and
the cutoff energy for the plane-wave basis set is 500 eV. The DFT iterations
were repeated until the total energy change was more than 10-8 eV between
electronic steps and more than 10-6 eV between ionic steps.

Calculations of the phonon dispersion curves for the high-temperature phase
($R\bar{3}m$) were performed using frozen-phonon technique implemented in
Phonopy code [29]. For this purpose the 3$\times$3$\times$3 supercell
constructed from the primitive one was used. For such a cell
6$\times$6$\times$6 mesh of Brillouin zone in the reciprocal space was chosen.
The experimental crystal structure was fully relaxed before lattice dynamics
simulations.

Phonon frequencies for the low-temperature structure (space group $Cm$) were
obtained only for the $\Gamma$-point of the Brillouin zone using density
functional perturbation theory (DFPT) [30]. For the crystal structure
suggested in [25] only atomic positions were optimized and 6$\times$6$\times$6
mesh of Brillouin zone in the reciprocal space was used.

Structural optimization for the considered stacking patterns in the low-
temperature structure (space groups $Cm$ and $Cc$) were performed with the
8$\times$6$\times$4 mesh of Brillouin zone in the reciprocal space.

##  III Experimental results

In order to obtain information about symmetry of the observed excitations,
polarization measurements were performed in two geometries, with parallel (XX)
and mutually perpendicular (XY) polarizations of the incident and scattered
light, where X and Y refer to the directions in the basal plane of crystals
that have a hexagonal shape. In case of $R\bar{3}m$ crystal structure XX
geometry is characterized by the $A_{1g}$ \+ $E_{g}$ symmetry, and XY - by the
$E_{g}$ symmetry. Thus, these two measurements make it possible to obtain
information about the behavior of pure symmetric components of the spectrum.

Figure 3:  Raman spectra of as-grown samples at 300K for both scattering
symmetries (low panel) and extracted symmetry-dependent contributions (upper
panel). The 532 nm laser was used.

The high-temperature (above 500 K) rhombohedral phase with $R\bar{3}m$ space
group has the $A_{1g}+2A_{2u}+2E_{u}+E_{g}$ optical phonons at the center of
the Brillouin zone. Two of them, $E_{g}$ and $A_{1g}$, are Raman active. They
are observed at 486 and 595 cm-1 in LiCoO2 [31], at 465 and 544 cm-1 in LiNiO2
[32], and at 456 and 589 cm-1 in LiCrO2 [33]. For the $A_{1g}$ mode, the
atomic shift of oxygen atoms is along the $c$-axis, whereas the $E_{g}$ mode
corresponds to vibrations in the basal plane.

Figure 4:  Temperature dependent Raman spectra in XX scattering geometry
measured in the heat-cool cycle of the first sample. Vertical line marks
energy position of intensive maximum before cycling.

Going through the structural transition our Raman measurements clearly detect
lowering of the symmetry in LiVO2. Raman spectra of two measured at 300K LiVO2
samples show a much larger number of phonon lines than those predicted by the
selection rules for the $R\bar{3}m$ structure (Fig. 3). Three intense lines
are observed in the region of 140-310 cm-1. Moreover, 8-9 low-intensity lines
in the XX and 3-4 modes in the XY spectra were measured in the region of
350-700 cm-1. Thus, low-frequency broad lines at 144 and 247 cm-1, as well as
lines at 406, 458, and 495 cm-1 are observed in both polarized (XX) and
depolarized (XY) geometries (low panel). This is impossible for previously
proposed $Cm$ monoclinic structure of the low-temperature phase [25], where
the Raman active modes are $A^{\prime}$ and $A^{\prime\prime}$ vibrations.
Thus, we can conclude that the symmetry of the low-temperature phase is not
lower than trigonal. After subtracting XY from XX spectrum, we obtained pure
$A_{1g}$ and $E_{g}$ spectra (upper panel of Fig. 3) under the assumption of
trigonal symmetry of the low-temperature phase.

A large number of lines observed at low temperatures can be ascribed to change
in the lattice symmetry, namely formation of V trimers previously suggested to
explain magnetic susceptibility data and recently observed by the PDF analysis
[17]. This leads to a tripling of the unit cell [15; 16; 24] and to possible
folding of the Brillouin zone. This results in additional active Raman modes
in the center of the new Brillouin zone. To confirm this assumption, Raman
spectra were measured while temperature cycling through the phase transition
point. The close attention will be paid to the low-frequency region of the
spectrum, since intense lines in this region are not expected in the Raman
spectra for the $R\bar{3}m$ structure.

To sum up this section, the measurements performed clearly detect a radical
restructuring of the phonon spectrum as a result of the structural transition.
These changes are not only related to the V trimerization in the low-
temperature phase, as we will discuss below, but the spectra can be rather
sensitive to thermal cycling featuring slow structural dynamics and different
degree of the stacking faults in different samples.

###  III.1 Thermal measurements

We performed thermal measurements on two seemingly similar single crystals. In
Fig. 4 we present Raman spectra measured at different temperatures in the XX
geometry. One can see that after heating the first sample to 540 K, the
spectrum measured in this geometry completely disappears. Upon subsequent
cooling down to 445 K it recovers, but the frequency of main peak turns out to
be lower than initial. Fig. 5 illustrates spectral changes under thermal
cycling for the both scattering symmetries. Characteristic feature of as-grown
sample is the hardening of broad (80-100 cm-1) lines near 142 and 245 cm-1 in
the $E_{g}$ spectrum as temperature increases. Finally these two modes almost
merge into an asymmetric line at 253 cm-1 at 520 K shortly before its
disappearance. This behavior of the $E_{g}$ modes with increasing temperature
contrasts to the softening observed for the $A_{1g}$ mode. After cooling to
300 K the $E_{g}$ spectrum contains two narrow lines at 256 and 279 cm-1,
which apparently indicates that ordering has occurred. At the same time, the
line in the $A_{1g}$ spectrum shows a softening by $\sim$20 cm-1 and
broadening compared to the original spectrum.

Figure 5:  Evolution of the $A_{1g}$ and $E_{g}$ Raman spectra at the first
temperature cycle.

These changes in Raman spectra after thermal cycling can evidence the slow
structural reconstruction, which likely engage modifications of the stacking
order. In order to study this effect further on we performed the second cycle
of thermal scanning through the magnetic transition temperature and the
spectrum disappeared after 520 K (540 K in the first run) with heating and it
recovered below 450 K with cooling (Fig. 6). After cooling, the energies of
the $E_{g}$ lines returned to their original values, remaining narrow, while
the frequency of the $A_{1g}$ line decreased and showed the presence of a
shoulder or sometimes two shoulders, depending on point of the sample, where
the measurement was performed.

Figure 6:  $A_{1g}$ and $E_{g}$ Raman spectra at the second temperature cycle.
Figure 7:  $A_{1g}$ and $E_{g}$ Raman spectra of the first and second samples
before and after 2 thermal cycles.

Results obtained during temperature cycling of the second sample are in
qualitative agreement with those obtained for the first crystal. The spectra
for both symmetries became identical and almost completely disappeared at 510K
(Fig. S1) [34]. When passing the temperature of magnetic transition, the
sample observed on the video camera showed a rather strong and sharp movement
in the basal plane of the crystal. A low-intensity broad band in the region of
210 cm-1 persisted at higher temperatures. When the sample was cooled below
465K, broad peaks at 260 cm-1 appeared in the spectra of both symmetries,
which acquired individual features upon further cooling.

However, the $A_{1g}$ and $E_{g}$ spectra measured at 300K were noticeably
different from the spectra obtained after thermal cycling on the first sample,
as can be seen from Fig. S2 and Fig. 7, which shows the spectra of both as-
grown samples before and after repeated thermal cycling. Despite the fact that
both samples were synthesized under the same conditions and their initial
spectra at 300 K are the same, the spectra of the second sample after thermal
cycling are very similar to the spectra of as-grown samples, while the spectra
of the first sample show a more ordered structure in both the low- and high-
frequency regions. Most likely, these differences are associated with
different ordering of the trimers in the direction perpendicular to the ab
plane. The second case may be due the trimer disorder in the stacking
direction while the first one obviously represents a well-ordered structure of
trimerized layers in the stacking direction. Although these results were
obtained from measurements at various points on both crystals, including both
natural and cleaved areas, the reason for this discrepancy is currently
unclear. It should be noted that despite the different degrees of ordering of
the trimers in the two crystals studied, the observed temperatures of the
structural transition are very close.

The temperature behavior of the low-frequency $E_{g}$ mode near 150 cm-1
turned out to be unusual, since its frequency increases when heated up to 500
K in all investigated crystals. But if in the first crystal (supposedly more
ordered) frequency of this mode practically did not change during the reverse
transition, then in the second crystal its frequency decreases from 210 to 144
cm-1 upon cooling from 465K to 300K and practically returned to its original
position, see Fig. S2. Upon further cooling down to 85 K it softens on another
40 cm-1, see Fig. S3 (“soft mode”?). Similar low-temperature results are
observed for the third crystal for various phonon symmetries (Fig. S4). The
observed temperature behavior suggests that this soft mode is a folded
counteract of one acoustic branch of the $R\bar{3}m$ phase, which becomes
unstable during the magnetic transition [25]. However, the softening slows
down at low temperatures, suggesting stability of the observed phase. It
should also be noted that this behavior is observed only in disordered sample
(sample 2), which indicates better stability of the first crystal with ordered
trimerized layers.

As for the $A_{1g}$ and $E_{g}$ phonons allowed for the $R\bar{3}m$ structure
at temperatures above the magnetic transition point, we observed a broad line
of the A1g symmetry near 580 cm-1, which significantly increases (up to 630
cm-1) upon cooling below the transition temperature. Such strong changes in a
rather narrow temperature range of 30 K are probably associated with a strong
decrease of the lattice constant a [24]. A broad band (100 cm-1) of the
$E_{g}$ symmetry is observed at high temperatures near 430 cm-1, but it is
difficult to attribute it definitely to any of the three narrow $E_{g}$ lines
that develop in this energy range when the crystal is cooled below the
transition temperature.

To sum up this section, the thermal cycling can reveal difference in two
seemingly similar LiVO2 single crystals, which is likely due to different
degree of stacking faults (our first sample is more ordered that the second
one).

###  III.2 Analysis of low-temperature data

The proposed cell of the low-temperature phase [16; 19] has a lattice
parameter $a^{*}=\sqrt{3}a$ and is rotated by 30∘ relative to the initial
(high-temperature $R\bar{3}m$) one. The wave vector (1/3,-1/3,0) for the
primitive $R\bar{3}m$ cell (in notations of the equivalent hexagonal Brillouin
zone (BZ)-the point K) is exactly 2 times larger than the wave vector of the
$M^{\prime}$ point on the boundary of the reduced BZ (Fig. 8). Consequently,
as a result of the BZ folding, the phonons from initial zone boundary at the
$K$ point may appear in the center of the new zone and become Raman active.
Thus, it can be assumed that 3 intense low-frequency lines are the result of
the folding of the acoustic $A_{2u}$ and $E_{u}$ branches in the new BZ, and
high-frequency lines are result of the folding of the $A_{1g}$, $A_{2u}$,
$E_{g}$, and $E_{u}$ optical branches. The experiment suggests that each of
the $E_{u}$ acoustic branches forms the $E_{g}$ doublet at the $\Gamma$ point,
so similar symmetry is expected when folding the 2 $E_{u}$ optical modes. In
all measurements, we observed an asymmetric $A_{1g}$ line (or two lines) at
280-300 cm-1, so it is natural to assume that the $A_{u}$ branches fold up
into $A_{1g}$ at the $\Gamma$ point. In such a scenario together with the
original optical modes at $\Gamma$ the phonon spectrum of the low-temperature
phase will contain at least 33 optical modes.

Figure 8:  Reciprocal space of the triangular lattice with black and red
hexagons denoting the high-temperature ($R\bar{3}m$) and possible low-
temperature Brillouin zones.

As can be seen from Fig. 7, at least 16 lines (8-9 A1g and 8-9 Eg) are
observed in the spectra and other may be nonactive in Raman. To confirm the
fact of phonon back-folding we tried to compare the obtained spectra with our
calculations of the phonon spectrum of the high-temperature phase. For this
comparison, we used the calculated for $R\bar{3}m$ phase frequencies at the
$\Gamma$-A direction. (Fig. 9).

Figure 9:  Calculated in non-magnetic GGA phonon dispersion curves for the
high-temperature ($R\bar{3}m$) phase of LiVO2. High-symmetry points of the
Brilluoin zone are taken for a primitive cell in rhombohedral axis. Absolute
minimum in the negative values in $\Gamma$-A direction corresponds to the
$\bf{q}$-vector $(\frac{1}{3},-\frac{1}{3},0)$ (further in the text the
corresponding point is named K after the point in cell with the hexagonal
axis).

The comparison presented in Tab. 1 shows some agreement (especially in the
high-frequency range) between the experimental and calculated frequencies,
which provides arguments in favor of observing folded phonons. The result is
quite striking, since the observation of folded phonons in a Raman spectrum is
an uncommon phenomenon. Such spectra are well known in the case of
measurements of different superlattices (e.g. various SiC polytypes [35] or
layered systems having charge density instabilities [36; 37]); sometimes new
lines were interpreted as phonons activated by the appearance of the magnetic
Brillouin zone as a result of a magnetic transition [38].

Although the exact crystal structure of the trimerized phase is unknown (most
probably because of the stacking faults) we compare the experimental and
calculated optical frequencies for the $Cm$ structure [25] (they are presented
in the last column of Tab. 1). Although experimental phonon symmetries do not
agree with those obtained in these calculations, the frequency range of the
observed and calculated energies agrees quite well.

Experiment | Calculations
---|---
No. | Symmetry | Mode | $R\bar{3}m$ | $Cm$
$\Gamma$-point | K-point |  $\Gamma$-point
1 | $E_{g}$ | 260 |  |  | 263
2 | $A_{1g}$ | 268 | 249 | 258 | 265
3 | $E_{g}$ | 281 |  |  | 277
4 | $A_{1g}$ | 287 |  |  | 279
5 | $E_{g}$ | 343 |  | 349 | 326, 327, 333, 349
6 | $E_{g}$ | 368 | 372 | 371, 374 | 372, 377
7 | $A_{1g}$ | 390 | 397 |  | 382, 394, 397
8 | $A_{1g}$ | 420 | 398 | 421 | 409, 410
9 | $E_{g}$ | 424 |  |  | 435
10 | $A_{1g}$ | 460 |  | 447 | 447, 456
11 | $E_{g}$ | 462 |  |  | 463, 464
12 | $E_{g}$ | 480 |  | 483 | 480, 481
13 | $E_{g}$ | 507 |  | 495, 500 | 509
14 | $E_{g}$ | 520 |  | 505 | 526
15 | $A_{1g}$ | 527 | 573 |  | 532
16 | $A_{1g}$ | 618 | 598, 599 |  | 592, 596
17 | $A_{1g}$ | 650 |  |  | 665, 704
Table 1: Comparison between experimentally found (for the first sample after
thermal cycling) and calculated for both high-temperature (at $\Gamma$ and at
K$=(\frac{1}{3},-\frac{1}{3},0)$) and low-temperature (at $\Gamma$) phases
phonon frequencies given in cm-1. Symmetry are given for the experimental
modes. The calculated Raman active modes in the $\Gamma$-point for the
$R\bar{3}m$ structure are given in bold.

##  IV Calculations of different types of stacking

In order to check the possibility of stacking faults in LiVO2 one can estimate
the total energy difference between the types of stacking. We started with the
structure suggested by Subedi in [25] (Fig. 10(a-b)) and doubled the primitive
cell along the $c$-axis in order to introduce corresponding atomic shifts. The
lower-lying layer of V atoms was untouched while the upper one was shifted by
either $b/3$ (Fig. 10(c-d)), or $2b/3$ (Fig. 10(e-f)). This immediately
changes symmetry of the structure to the $Cc$ space group.

Figure 10:  Considered types of stacking in LiVO2 along the $c$-axis. Only
vanadium ions in the doubled primitive cell are shown. V in the nearest layers
are given in different colors (blue for first layer in the cell and red for
the upper one). (a-b) Initial structure suggested in [25] (space group $Cm$),
(c-d) the structure with the $b/3$ and (e-f) $2b/3$ shift for the nearest V
layer (in comparison to the initial one).

We performed structural optimization (in non-magnetic GGA) for all three
stacking types and all the initial patterns were retained in the resulting
structures demonstrating their stability. The total energies of different
stacking types differ by a tiny amount of 0.1 meV/f.u. Thus, one could
naturally expect stacking faults in LiVO2, which explains not only previous
X-ray experiments, but also our Raman measurements, when two seemingly similar
samples show a different spectra under thermal cycling.

##  V Conclusions

In the present paper we demonstrate that Raman spectroscopy is a sensitive
probe helpful in studying a long-standing problem of describing the structural
and magnetic transition observed in LiVO2 at $\sim$ 500K. It clearly shows
that the symmetry of a low-temperature phase supposedly having trimerized
crystal structure (V trimers) is not lower than trigonal. Detailed analysis of
Raman spectra of single crystals under thermal cycling evidence a slow
structural dynamics (e.g. slow rearranging of V trimers) or stacking faults.
The degree of stacking faults can be different in seemingly similar samples.
Thus, the Raman spectroscopy can be used as a hallmark for testing LiVO2
samples in further studies. Theoretical calculations based on the density
functional theory supports possible stacking faults demonstrating that the
energy of different stacking types is of order 1 K per formula unit.

##  VI Acknowledgments

The authors are grateful to A.V. Ushakov and R. Maezono for useful discussions
and I.V. Baklanova for help with low-temperature measurements. The research
was carried out within the state assignment of Ministry of Science and Higher
Education of the Russian Federation (themes “Electron” No 122021000039-4 and
“Quantum” No 122021000038-7). A portion of this work was supported by National
Key R&D Program of China (No. 2023YFA1406100).

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