# Driving Mechanism and Dynamic Fluctuations of Charge Density Waves in the
Kagome Metal ScV6Sn6

Shuyuan Liu1,2, Chongze Wang2, Shichang Yao2, Yu Jia1,3, Zhenyu Zhang4∗, and
Jun-Hyung Cho2∗ 1Joint Center for Theoretical Physics, School of Physics and
Electronics, Henan University, Kaifeng 475004, People’s Republic of China
2Department of Physics and Research Institute for Natural Science, Hanyang
University, 222 Wangsimni-ro, Seongdong-Ku, Seoul 04763, Republic of Korea
3Key Laboratory for Special Functional Materials of the Ministry of Education,
Henan University, Kaifeng 475004, People’s Republic of China
4International Center for Quantum Design of Functional Materials (ICQD), Hefei
National Laboratory for Physical Sciences at Microscale, and Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei 230026, China

###### Abstract

In contrast to the AV3Sb5 (A = K, Rb, Cs) family exhibiting the ubiquitous
2${\times}$2${\times}$2 CDW order, only ScV6Sn6 in the RV6Sn6 (R = Sc, Y, La)
family displays the unusual CDW with a $\sqrt{3}{\times}\sqrt{3}$ in-plane
ordering and a tripling of the unit cell along the $c$-axis. Here, using
first-principles density-functional theory calculations, we show that both the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 and $\sqrt{3}{\times}\sqrt{3}{\times}$3
CDW orderings can be driven by a Jahn-Teller-like effect where the Sn atoms
residing in the kagome bilayers partially undergo delicately different schemes
of interlayer dimerization, accompanied by charge redistribution between such
Sn atoms and band gap opening. Counterintuitively, whereas the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 phase is energetically more stable than
$\sqrt{3}{\times}\sqrt{3}{\times}$3, the latter is thermodynamically
stabilized above the CDW transition temperature $T_{\rm CDW}$ by its higher
configurational entropy contributed by degenerate fluctuating phases, and is
kinetically selected below $T_{\rm CDW}$, as corroborated by experimental
observation of the accompanying first-order phase transition. Our findings
reveal the order-disorder nature of the $\sqrt{3}{\times}\sqrt{3}{\times}$3
CDW phase transition in ScV6Sn6, with broader implications for understanding
charge orderings and CDW fluctuations in other kagome metals.

$Introduction$.$-$ The two-dimensional (2D) kagome lattice consisting of
corner-sharing triangles has attracted tremendous attention due to its unique
electronic structure involving a flat band, Dirac cone, and van Hove
singularities (VHSs) [1, 2, 3]. So far, many experimental and theoretical
works in different kagome compounds have presented a wide variety of novel
quantum phenomena like anomalous Hall effect [4, 5, 6], spin liquid [7, 8],
flatband ferromagnetism [9], charge density wave (CDW) [10, 11], and
superconductivity [12, 13, 14]. Specifically, a family of V-based kagome
metals AV3Sb5 (A = K, Rb, Cs) possesses nontrivial band topology and
correlated many-body states such as the unconventional CDW order breaking both
time-reversal and rotational symmetries [15, 16], electronic nematicity [17,
18], and superconductivity [19, 20, 21, 22]. Considering that the AV3Sb5
compounds have the saddle points of linearly dispersive Dirac bands at three
inequivalent $M$ points near the Fermi level $E_{F}$, the nesting of such VHSs
was proposed to be a driving mechanism for the CDW formation via electron-
electron interaction [23, 24, 25, 26]. However, the origin of CDW in AV3Sb5 is
still under debate, together with momentum-dependent electron-phonon coupling
or phonon softening [27, 28, 29, 30, 31]. To resolve this debate, more
extensive studies of the CDW formation in other V-based kagome metals are
highly desired.

Recently, among the new nonmagnetic RV6Sn6 (R = Sc, Y, La) family consisting
of V kagome bilayers [see Fig. 1(a)], only ScV6Sn6 undergoes a first-order CDW
transition at $T_{\rm CDW}$ $\approx$ 92 K [32]. Here, ScV6Sn6 was
experimentally observed to exhibit a $\sqrt{3}$${\times}$$\sqrt{3}$${\times}$3
CDW order, which is distinct from the 2${\times}$2${\times}$2 CDW order in
AV3Sb5 [10, 33]. Such entirely different wave lengths and orientations of the
CDW orders between ScV6Sn6 and AV3Sb5 may provide a new route to gain insights
into the origin of the CDW order in V-based kagome metals. Specifically, while
the CDW order in AV3Sb5 accompanies large in-plane displacements of the V
atoms [33], that in ScV6Sn6 is dominated by displacements of the Sc and Sn
atoms along the $c$-axis [34]. These unusual features of the CDW order in
ScV6Sn6 challenge the nesting scenario because the scattering between the
$M$-saddle points near $E_{F}$ cannot be responsible for a driving force of
the $\sqrt{3}$${\times}$$\sqrt{3}$${\times}$3 periodic lattice distortion.

For ScV6Sn6, diffuse scattering (DS) and inelastic x-ray scattering (IXS)
experiments [35] reported the presence of an order-disorder CDW phase
transition at $T_{\rm CDW}$, where the disordered phase containing a short-
range $\sqrt{3}$${\times}$$\sqrt{3}$${\times}$2 CDW order was suppressed and
replaced by a long-range $\sqrt{3}$${\times}$$\sqrt{3}$${\times}$3 CDW order.
Another DS and IXS experiments [36] observed that the low energy longitudinal
phonon with propagation wavevector $q^{*}$ = (${1\over 3}$, ${1\over 3}$,
${1\over 2}$) collapses at ${\sim}$98 K due to the electron-phonon
interaction. Here, the collapse of a soft mode at $q^{*}$ is driven by a
softening of a flat phonon plane at $k_{z}$ = ${\pi}$, characterized by an
out-of-plane vibration of the trigonal Sn atoms. However, the long-range
charge order does not emerge at $q^{*}$ but at a different wavevector $q_{s}$
= (${1\over 3}$, ${1\over 3}$, ${1\over 3}$). It is interesting to compare the
characteristics of order-disorder CDW phase transition between ScV6Sn6 and
AV3Sb5. For AV3Sb5, recent DS and IXS experiments [37, 38, 39] also reported a
first-order CDW phase transition that features an order-disorder
transformation type with strong CDW fluctuations above $T_{\rm CDW}$. Note
that inelastic x-ray, neutron, and Raman scattering experiments for AV3Sb5
[37, 38, 39] have observed the absence of the lattice collapse at $T_{\rm
CDW}$. Therefore, both ScV6Sn6 [35, 40] and AV3Sb5 [37, 38, 39] undergo the
first-order, order-disorder CDW phase transition, but their characteristics
are apparently distinct from each other: i.e., ScV6Sn6 features a first-order
transition with phonon softening while AV3Sb5 exhibits CDW fluctuations with
the absence of phonon softening. Specifically, the temperature evolution of
the integrated IXS intensities in ScV6Sn6 [35] showed that the maximum peak
intensity of the $\sqrt{3}{\times}\sqrt{3}{\times}$3 CDW order below $T_{\rm
CDW}$ is at least 3 orders of magnitude larger than that of the short-range
$\sqrt{3}{\times}\sqrt{3}{\times}$2 CDW order above $T_{\rm CDW}$. This huge
different IXS intensities of two CDW orders together with the inconsistency
between the soft $q^{*}$-phonon and $q_{s}$-CDW wavevectors are puzzling.
Microscopic understanding of such an unconventional CDW phase transition in
ScV6Sn6 is yet to be identified.

Figure 1: Atomic structure of pristine and CDW phases of ScV6Sn6. (a)
Optimized structure of the 1${\times}$1${\times}$1 pristine phase of ScV6Sn6,
together with its Brillouin zone. The top/side views of Sn dimers in the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 and $\sqrt{3}{\times}\sqrt{3}{\times}$3
phases are drawn in (b). The black dashed lines in (b) represent the hexagonal
unit cells, while the red dashed line in the
$\sqrt{3}{\times}\sqrt{3}{\times}$3 phase represents its primitive
rhombohedral unit cell. The numbers represent the Sn1-Sn1 bond lengths in
angstroms.

In this Letter, we investigate the driving mechanism of CDW and the nature of
CDW phase transition in ScV6Sn6 using first-principles density-functional
theory (DFT) calculations. We demonstrate that both the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 and $\sqrt{3}{\times}\sqrt{3}{\times}$3
CDW orderings can be driven by a Jahn-Teller-like effect where the Sn atoms
residing in kagome bilayers are partially dimerized along the $c$-axis
together with the charge redistribution between such Sn atoms and its
associated band gap opening. The $\sqrt{3}{\times}\sqrt{3}{\times}$3 phase is
found to be energetically less stable than
$\sqrt{3}{\times}\sqrt{3}{\times}$2, but the former becomes thermodynamically
stabilized above $T_{\rm CDW}$ due to its higher vibrational, electronic, and
configurational entropies compared to the latter. Furthermore, we reveal that,
while the $\sqrt{3}{\times}\sqrt{3}{\times}$3 CDW order shows dynamic
fluctuations between its degenerate configurations above $T_{\rm CDW}$, it is
kinetically trapped below $T_{\rm CDW}$, thereby leading to the first-order,
order-disorder phase transition with a release of configurational entropy at
$T_{\rm CDW}$. Therefore, we conclude that the majority of disordered phase
above $T_{\rm CDW}$ is composed of the dynamically fluctuating
$\sqrt{3}$${\times}$$\sqrt{3}$${\times}$3 CDW order, while the minority is a
local ordering of the short-range $\sqrt{3}$${\times}$$\sqrt{3}$${\times}$2
CDW order due to a relatively higher energy barrier between its degenerate
configurations, as discussed below.

$Results$.$-$ We begin by optimizing the atomic structure of the
1${\times}$1${\times}$1 pristine phase of ScV6Sn6 using the DFT scheme [41].
Figure 1(a) shows the optimized structure of the pristine phase with the
symmetry of space group $P6/mmm$, which is composed of the two V3Sn1 kagome
layers with the triangular Sn1 sublattice centered on the V hexagons of each
kagome lattice and the Sn2 honeycomb or Sn3/Sc honeycomb/triangular layer
between neighboring V3Sn1 kagome layers. The electronic band structure of this
pristine phase is displayed in Fig. 2(a), together with its projection onto
the V 3$d$ and Sn 5$p$ orbitals (see also Fig. S1 in the Supplemental Material
[51]). We find that the bands with momentum-dependent orbital characters near
$E_{F}$ agree well with angle-resolved photoemission spectroscopy (ARPES) and
optical reflection measurements [52, 53, 54]: i.e., the states along the
${\Gamma}$-$M$-$K$-${\Gamma}$ line originate from all the five V $d$ orbitals,
while those along the $A$-$L$-$H$-$A$ line mainly originate from the Sn1
$p_{z}$ and V $d_{xz,yz}$ orbitals. The latter hybridized states in the V3Sn1
kagome layers play a crucial role in the CDW formation, as discussed below.
Meanwhile, two saddle points at the $M$ point close to $E_{F}$ (indicated by
red arrows) don’t produce the corresponding VHSs in the partial density of
states (PDOS) [see Fig. 2(a)]. Therefore, we can say that the Fermi-surface
nesting of saddle-point derived VHSs between three distinct $M$ points [55],
previously proposed in AV3Sb5 [23, 24, 25, 26], is not responsible for the
driving mechanism of the CDW order in ScV6Sn6 [34]. Indeed, the electronic
states along the ${\Gamma}$-$M$-$K$-${\Gamma}$ line remain intact across the
CDW transition, as will be shown later.

Figure 2: Electronic structure of pristine and CDW phases of ScV6Sn6.
Calculated band structures of the (a) pristine and (b)
$\sqrt{3}{\times}\sqrt{3}{\times}$2 phases of ScV6Sn6. The bands are projected
onto V 3$d$ and Sn 5$p$ orbitals, where the radii of circles are proportional
to the weights of the corresponding orbitals. In (a), the PDOS and two saddle
points at the $M$ point (indicated by arrows in the inset) are included. In
(b), the charge characters of the $S_{1}$ and $S_{2}$ states at the
$H^{\prime}$ point are displayed with an isosurface of 8${\times}10^{-3}e$/Å3.
In (c), the pristine bands along the three ${\alpha}$-${\beta}$ paths (one
$P_{1}$ and two $P_{2}$) are folded to form the six-fold degenerate state
$S_{0}$ at $H^{\prime}$, the charge character of which is also displayed with
an isosurface of 8${\times}10^{-3}e$/Å3.

To explore the phonon instability of the pristine structure of ScV6Sn6, we
calculate its phonon spectrum using the frozen-phonon method [47, 48]. As
shown in Fig. S4(a), there exist imaginary phonon modes along the entire
$A$-$L$-$H$-$A$ line, the lowest one of which is located at the $H$ =
(${1\over 3}$, ${1\over 3}$, ${1\over 2}$) point, leading to a structural
reconstruction within the $\sqrt{3}{\times}\sqrt{3}{\times}$2 unit cell. Here,
the $H$-phonon mode represents the Sn1-Sn1 bond stretching vibrations along
the c-axis [see Fig. S4(b)]. The resulting $\sqrt{3}{\times}\sqrt{3}{\times}$2
CDW order with the $D_{6h}$ symmetry of space group $P6/mmm$ becomes more
stable than the pristine phase by 12.59 meV per formula unit (f.u.). In the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 CDW structure, the Sn1-Sn1 bond length
$d_{\rm dimer}$ [see Fig. 1(b)] is shortened (lengthened) to be 3.052 or 3.073
Å (3.710 or 3.740 Å) compared to that (3.339 Å) in the pristine structure.
Hereafter, two Sn1 atoms between neighboring V3Sn1 kagome layers are termed Sn
dimers. While the CDW structure of AV3Sb5 has the large in-plane displacement
of V kagome sublattice forming trimers and hexamers [33], that of ScV6Sn6 has
the minimal displacement of V atoms with the large out-of-plane displacement
of Sc and Sn atoms [34]. The variations of $d_{\rm dimer}$ during the CDW
formation of ScV6Sn6 implies a charge transfer from the longer to the shorter
Sn dimers, which in turn opens partial CDW gaps around the $H^{\prime}$ =
(${1\over 3}$, 0, ${1\over 4}$) point [see Fig. 2(b)]. Note that (i) the
folded pristine band structure onto the $\sqrt{3}{\times}\sqrt{3}{\times}$2
supercell shows that the hybridized Sn1 $p_{z}$ and V $d_{xz,yz}$ states
[$S_{0}$ in Fig. 2(c)] at $H^{\prime}$ on the boundary of Brillouin zone have
the six-fold degeneracy, which is split into the three occupied and three
unoccupied states in the $\sqrt{3}{\times}\sqrt{3}{\times}$2 phase [see Fig.
2(b)] and (ii) the unfolded band structure of the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 phase shows little change along the
${\Gamma}$-$M$-$K$-${\Gamma}$ line around $E_{F}$ (see Fig. S5). Here, the
six-fold degeneracy at $H^{\prime}$ is generated by the folding of pristine
bands along three different paths [see Fig. 2(c)]. As shown in Figs. 2(b) and
2(c), the charge character of the highest (lowest) occupied $S_{1}$
(unoccupied $S_{2}$) state at $H^{\prime}$ represents the charge accumulation
(depletion) in the shorter (longer) Sn1-Sn1 bonds, compared to the $S_{0}$
state in the pristine phase. Therefore, the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 CDW order can be attributed to a Jahn-
Teller-like effect involving the Sn1-Sn1 bond distortions, the charge
redistribution between such Sn dimers, and the resultant partial gap opening
around the $H^{\prime}$ point.

Table I shows the comparison of the lattice constants and $d_{\rm dimer}$ in
the pristine phase of RV6Sn6 (R = Sc, Y, La). We find that the lattice
constants $a$ = $b$ and $c$ in YV6Sn6 (LaV6Sn6) are larger than those in
ScV6Sn6 by 0.84(1.91)% and 0.14(0.63)%, respectively. Meanwhile, YV6Sn6
(LaV6Sn6) has a much shorter $d_{\rm dimer}$ value of 3.151(3.051) Å, compared
to that (3.339 Å) in ScV6Sn6. Note that the $d_{\rm dimer}$ values in the
former compounds are close to the shortest one (3.052 Å) in the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 CDW phase of ScV6Sn6. Therefore, we can
say that the Jahn-Teller-like CDW formation with the Sn1-Sn1 bond distortions
and their charge redistribution would not be necessary in YV6Sn6 and LaV6Sn6.
According to the magnetization and x-ray diffraction measurements [56] of Y-
and Lu-doped ScV6Sn6 where Sc is substituted by larger Y or Lu, the CDW order
was observed to be suppressed because the Sn1-Sn1 bond length is reduced by Y
and Lu doping. It is thus expected that external perturbations such as
chemical doping and pressure in RV6Sn6 can tune the Sn1-Sn1 bond length to
suppress or create CDW orders.

Table 1: Calculated lattice constants and $d_{\rm dimer}$ in the pristine phase of RV6Sn6 (R = Sc, Y, La). |  $a$ = $b$ (Å) |  $c$ (Å) |  $d_{\rm dimer}$ (Å)
---|---|---|---
ScV6Sn6 | 5.453 | 9.235 | 3.339
YV6Sn6 | 5.499 | 9.248 | 3.151
LaV6Sn6 | 5.557 | 9.293 | 3.051

Unlike the DFT prediction of the $\sqrt{3}{\times}\sqrt{3}{\times}$2 CDW
ground state [34], experimental evidence [32, 57, 58] showed that ScV6Sn6
undergoes a CDW transition to the $\sqrt{3}{\times}\sqrt{3}{\times}$3 phase at
$T_{\rm CDW}{\approx}$92 K. Therefore, the temperature versus free energy
landscape for the $\sqrt{3}{\times}\sqrt{3}{\times}$2 and
$\sqrt{3}{\times}\sqrt{3}{\times}$3 phases can be illustrated like Fig. 3(a):
i.e., the free energy of the $\sqrt{3}{\times}\sqrt{3}{\times}$3 phase becomes
lower than that of the $\sqrt{3}{\times}\sqrt{3}{\times}$2 phase above $T_{\rm
CDW}$. Our DFT calculations show that at zero temperature, the total energy of
the $\sqrt{3}{\times}\sqrt{3}{\times}$3 phase is higher than that of the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 phase by ${\Delta}E$ = 6.29 meV/f.u. The
$\sqrt{3}{\times}\sqrt{3}{\times}$3 CDW structure has two different kinds of
the Sn1-Sn1 bonds with $d_{\rm dimer}$ = 3.169 and 3.677 Å [see Fig. 1(b)].
Meanwhile, the band structure of the $\sqrt{3}{\times}\sqrt{3}{\times}$3 phase
is overall the same as that of the $\sqrt{3}{\times}\sqrt{3}{\times}$2 phase
(see Fig. S5), but the former has a CDW gap of 309 meV at the
$H^{\prime\prime}$ = (${1\over 3}$, 0, ${1\over 3}$) point, slightly smaller
than that (440 meV at $H^{\prime}$) in the latter.

Figure 3: (a) Schematic free energy landscape of the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 and $\sqrt{3}{\times}\sqrt{3}{\times}$3
phases at different temperatures and the energy profiles along the transition
pathways (b) between degenerate $\sqrt{3}{\times}\sqrt{3}{\times}$3
configurations, (c) between degenerate $\sqrt{3}{\times}\sqrt{3}{\times}$2
configurations, and (d) between the $\sqrt{3}{\times}\sqrt{3}{\times}$2 and
$\sqrt{3}{\times}\sqrt{3}{\times}$3 phases. The $E_{b}$ values in (b), (c),
and (d) are estimated as 6.30, 12.59, and 10.24 meV/f.u., respectively.

To estimate $T_{\rm CDW}$, we calculate the Helmholtz free energy difference
${\Delta}F$ = ${\Delta}E$ \+ ${\Delta}E_{\rm vib}$ \- $T{\Delta}S$ between the
$\sqrt{3}{\times}\sqrt{3}{\times}$3 and $\sqrt{3}{\times}\sqrt{3}{\times}$2
phases where $E_{\rm vib}$ represents vibrational energy [59] and $S$ includes
three different types of entropy such as vibrational entropy $S_{\rm vib}$,
electronic entropy $S_{\rm el}$, and configurational entropy $S_{\rm con}$.
$S_{\rm vib}$ is calculated from the phononic DOS $g^{\rm ph}(\epsilon)$ [47,
48]:

$S_{\rm vib}=-3k_{B}\int g^{\rm ph}(\epsilon)(n(\epsilon){\rm
ln}[n(\epsilon)]-[1+n(\epsilon)]{\rm ln}[1+n(\epsilon)])d\epsilon,$ (1)

where $k_{B}$ is the Boltzmann constant and $n(\epsilon)$ is the Bose-Einstein
population of a state of energy ${\epsilon}$ at $T$. Figure 4(a) displays
${\Delta}S_{\rm vib}$ = $S_{\rm vib,\sqrt{3}{\times}\sqrt{3}{\times}3}$ \-
$S_{\rm vib,\sqrt{3}{\times}\sqrt{3}{\times}2}$ as a function of $T$,
indicating that the $\sqrt{3}{\times}\sqrt{3}{\times}$3 phase has larger
$S_{\rm vib}$ than the $\sqrt{3}{\times}\sqrt{3}{\times}$2 phase. Therefore,
$T{\Delta}S_{\rm vib}$ increases with increasing $T$ (see Fig. S6), leading to
a decrease in ${\Delta}F$. Next, we calculate $S_{\rm el}$ from the electronic
DOS $g^{\rm el}(\epsilon)$:

$S_{\rm el}=-k_{B}\int g^{\rm el}(\epsilon)(n(\epsilon){\rm
ln}[n(\epsilon)]+[1-n(\epsilon)]{\rm ln}[1-n(\epsilon)])d\epsilon,$ (2)

where $n(\epsilon)$ is the Fermi-Dirac distribution function. Since the
$\sqrt{3}{\times}\sqrt{3}{\times}$3 phase has higher $g^{\rm el}(E_{F})$ than
the $\sqrt{3}{\times}\sqrt{3}{\times}$2 phase (see Fig. S7), the former has
larger $S_{\rm el}$ than the latter. However, due to the much smaller values
of $S_{\rm el}$ [see Fig. 4(a)], $T{\Delta}S_{\rm el}$ plays a minor role in
lowering ${\Delta}F$. Finally, we consider $S_{\rm con}$ arising from the
number of degenerate configurations to which the system can be accessible.
Using the nudged elastic band method [49, 50], we obtain an energy barrier
$E_{b}$ = 6.30 meV/f.u. along the transition pathway TP1 between degenerate
configurations in the $\sqrt{3}{\times}\sqrt{3}{\times}$3 structure [see Fig.
3(b)], leading to strong CDW fluctuations above $T_{\rm CDW}$, as discussed
below. Consequently, the $\sqrt{3}{\times}\sqrt{3}{\times}$3 phase can attain
a configuration entropy of $S_{\rm con}$ = $k_{B}$ln(6)/3 = 0.051
meV$\cdot$K-1/f.u. above $T_{\rm CDW}$ because there are six degenerate
configurations within the primitive unit cell containing three formula units
(see Fig. S8). On the other hand, the $\sqrt{3}{\times}\sqrt{3}{\times}$2
phase has relatively higher values of energy barrier [see Fig. 3(c)] and free
energy [see Fig. 3(a)] above $T_{\rm CDW}$ compared to the
$\sqrt{3}{\times}\sqrt{3}{\times}$3 phase, leading to $S_{\rm con}$ = 0
without dynamic fluctuations between degenerate configurations. Note that DS
and IXS experiments [35, 40] observed the $\sqrt{3}{\times}\sqrt{3}{\times}$2
phase above $T_{\rm CDW}$ as a local short-range ordering. Given the estimated
values of ${\Delta}E$, ${\Delta}E_{\rm vib}$, $T{\Delta}S_{\rm vib}$,
$T{\Delta}S_{\rm el}$, $T{\Delta}S_{\rm con}$, we find that the $F$ of the
high-entropy $\sqrt{3}{\times}\sqrt{3}{\times}$3 phase becomes equal to that
of the $\sqrt{3}{\times}\sqrt{3}{\times}$2 phase at ${\sim}$97 K, where
$T{\Delta}S_{\rm vib}$ = 1.03 meV/f.u., $T{\Delta}S_{\rm el}$ = 0.10 meV/f.u.,
$T{\Delta}S_{\rm con}$ = 4.99 meV/f.u., and ${\Delta}E_{\rm vib}$ = -0.17
meV/f.u. [59] are obtained. This predicted $T_{\rm CDW}$ value is close to the
experimental value of ${\sim}$92 K [32] [see Fig. 4(b)], with highlighting the
dominant role of configurational entropy in the
$\sqrt{3}{\times}\sqrt{3}{\times}$3 CDW formation.

Figure 4: Calculated ${\Delta}S$ and ${\Delta}F$. (a) ${\Delta}S_{\rm vib}$,
${\Delta}S_{\rm el}$, ${\Delta}S_{\rm con}$, and (b) ${\Delta}F$ as a function
of $T$. In (b), the calculated values of ${\Delta}E$+${\Delta}E_{\rm vib}$ and
$T{\Delta}S$ are also included.

According to the Arrhenius equation [60], the transition rate $R$ can be
estimated as $R$ = $\nu$ exp($-{E_{b}\over k_{B}T}$), where $\nu$ is the
attempt frequency and $E_{b}$ is the activation barrier. To estimate $R$ along
TP1, we adopt $\nu$ with the frequency (${\sim}$1.4 THz) of the Sn1-Sn1 bond
stretching phonon modes around the $H$ point in the
$\sqrt{3}{\times}\sqrt{3}{\times}$3 phase [see Fig. S4(c)]. With $E_{b}$ =
6.30 meV along TP1, we obtain $R$ = 7.61$\times$1011 s-1 at 120 K above
$T_{\rm CDW}$, indicating strong CDW fluctuations among the degenerate
configurations of the $\sqrt{3}{\times}\sqrt{3}{\times}$3 phase with a
characteristic lifetime of 0.91 ps [61]. Interestingly, this fluctuating CDW
lifetime is comparable with that (${\sim}$0.5 ps) measured by ultrafast
spectroscopy in cuprates [63]. We note that such CDW fluctuations above
$T_{\rm CDW}$ are similar to the case of CsV3Sb5 where the dynamic
fluctuations of the 2${\times}$2${\times}$2 CDW order was experimentally
observed above $T_{\rm CDW}$ = 94 K up to ${\sim}$160 K [39]. Just below
$T_{\rm CDW}$, the energy barrier from the $\sqrt{3}{\times}\sqrt{3}{\times}$3
to the $\sqrt{3}{\times}\sqrt{3}{\times}$2 phase is estimated as ${\sim}$10.24
meV [see Figs. 3(a) and 3(d)], relatively higher than that (6.30 meV) along
TP1. Therefore, the long-range $\sqrt{3}{\times}\sqrt{3}{\times}$3 phase can
be kinetically trapped in certain temperature ranges below $T_{\rm CDW}$
without undergoing a transition to the $\sqrt{3}{\times}\sqrt{3}{\times}$2
phase, which is supported by experimental evidence that the
$\sqrt{3}{\times}\sqrt{3}{\times}$3 phase was observed down to a temperature
of ${\sim}$50 K [32]. Furthermore, as shown in Fig. 4(b), a discontinuous
change in ${\Delta}F$ with a sizable release of configurational entropy
indicates the first-order character of CDW phase transition, consistent with
the presence of a sharp peak in the measured specific heat data [32] at
$T_{\rm CDW}$. It is worth noting that, as $T$ further decreases, both the
energy barrier from the $\sqrt{3}{\times}\sqrt{3}{\times}$3 to the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 phase and its thermal activation rate are
simultaneously lowered [see Fig. 3(a)]. Consequently, these two factors
represent the compensation effect that determines whether the
$\sqrt{3}{\times}\sqrt{3}{\times}$3 phase remains kinetically trapped or is
transformed into the $\sqrt{3}{\times}\sqrt{3}{\times}$2 phase. Indeed,
synchrotron x-ray diffraction experiments [40] observed the coexistence of the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 and $\sqrt{3}{\times}\sqrt{3}{\times}$3
charge correlations down to 15 K, indicating that the measured samples contain
a mixture of short-range $\sqrt{3}{\times}\sqrt{3}{\times}$2 and long-range
$\sqrt{3}{\times}\sqrt{3}{\times}$3 CDW orders at lower temperatures.

$Discussion$.$-$ Recently, IXS measurements [35] for ScV6Sn6 reported an
unusual CDW formation process with two competing CDW instabilities: i.e., the
long-range $\sqrt{3}{\times}\sqrt{3}{\times}$3 CDW below $T_{\rm CDW}$ while
the short-range $\sqrt{3}{\times}\sqrt{3}{\times}$2 CDW above $T_{\rm CDW}$.
Across the CDW phase transition, the growth of an elastic central peak
corresponding to the $\sqrt{3}{\times}\sqrt{3}{\times}$3 CDW was observed
together with phonon softening associated with the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 CDW. It is, however, a puzzle that the
soft phonon occurring at $q^{*}$ = (${1\over 3}$, ${1\over 3}$, ${1\over 2}$)
is accompanied by the CDW formation at a different wavevector $q_{s}$ =
(${1\over 3}$, ${1\over 3}$, ${1\over 3}$). Furthermore, the maximum IXS peak
intensity of the $\sqrt{3}{\times}\sqrt{3}{\times}$3 CDW is at least 3 orders
of magnitude larger than that of the $\sqrt{3}{\times}\sqrt{3}{\times}$2 CDW
[35], implying that the former phase has significantly larger domain sizes in
the measured samples than the latter one. Therefore, it is unlikely that there
exists a first-order transformation from the short-range
$\sqrt{3}{\times}\sqrt{3}{\times}$2 to the long-range
$\sqrt{3}{\times}\sqrt{3}{\times}$3 CDW orders, driven by the scenario of a
momentum-dependent electron-phonon coupling favoring $q_{s}$-CDW as the ground
state [35]. Instead, we believe that the observed first-order transition is of
an order-disorder type between the static $\sqrt{3}{\times}\sqrt{3}{\times}$3
CDW and its fluctuations, as discussed above. Here, the dynamic fluctuations
of $\sqrt{3}{\times}\sqrt{3}{\times}$3 CDW above $T_{\rm CDW}$ should sharply
weaken the intensity of elastic central peak at $q_{s}$, as measured by IXS
[35]. Meanwhile, according to synchrotron x-ray diffraction measurements [40],
the short-ranged charge correlations of the
$\sqrt{3}{\times}\sqrt{3}{\times}$2 phase was observed in a wide range of
temperatures between 15 and 300 K. This may be explained in terms of the local
ordering character of the $\sqrt{3}{\times}\sqrt{3}{\times}$2 phase due to a
relatively higher energy barrier between its degenerate configurations [see
Fig. 3(c)]. More refined experimental identifications of the phase-dependent
CDW fluctuations above $T_{\rm CDW}$ are needed in the future.

In conclusion, based on first-principles calculations, we have demonstrated
that the CDW order in ScV6Sn6 is driven by a Jahn-Teller-like effect involving
the Sn1-Sn1 bond distortions, charge redistribution, and partial band gap
opening. Furthermore, our free energy analysis revealed that the
configurational entropy in the $\sqrt{3}{\times}\sqrt{3}{\times}$3 phase
emerges above $T_{\rm CDW}$ due to CDW fluctuations, while below $T_{\rm
CDW}$, the long-range $\sqrt{3}{\times}\sqrt{3}{\times}$3 CDW order is
kinetically trapped, indicating a first-order, order-disorder phase
transition. Our findings clarify the driving mechanism of CDW and the order-
disorder nature of CDW phase transition in ScV6Sn6, which have important
implications for understanding the observed first-order transition with CDW
fluctuations in other kagome metal compounds such as AV3Sb5 [37, 38, 39] and
FeGe [64].

Acknowledgements. This work was supported by the National Research Foundation
of Korea (NRF) grant funded by the Korean Government (Grants No.
2022R1A2C1005456 and No. RS202300218998) and by BrainLink program funded by
the Ministry of Science and ICT through the National Research Foundation of
Korea (2022H1D3A3A01077468). Z.Z. acknowledges Innovation Program for Quantum
Science and Technology (Grant No. 2021ZD0302800) and Y.J. acknowledges
National Natural Science Foundation of China (Grant No. 12074099). The
calculations were performed by the KISTI Supercomputing Center through the
Strategic Support Program (Program No. KSC-2023-CRE-0112) for the
supercomputing application research.

S. L. and C. W. contributed equally to this work.

∗ Corresponding authors<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
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