# Raman Spectroscopic Study on Bi2Rh3Se2: Two-dimensional-Ising Charge Density
Wave and Quantum Fluctuations

Fei Jiao  Anhui Key Laboratory of Condensed Matter Physics at Extreme
Conditions, High Magnetic Field Laboratory,Chinese Academy of Sciences, Hefei
230031, China  Science Island Branch of Graduate School, University of Science
and Technology of China, Hefei 230026, China  Yonghui Zhou  Anhui Key
Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic
Field Laboratory,Chinese Academy of Sciences, Hefei 230031, China  Science
Island Branch of Graduate School, University of Science and Technology of
China, Hefei 230026, China  Shuyang Wang  Anhui Key Laboratory of Condensed
Matter Physics at Extreme Conditions, High Magnetic Field Laboratory,Chinese
Academy of Sciences, Hefei 230031, China  Chao An  Institutes of Physical
Science and Information Technology, Anhui University, Hefei 230601, China
Xuliang Chen  Anhui Key Laboratory of Condensed Matter Physics at Extreme
Conditions, High Magnetic Field Laboratory,Chinese Academy of Sciences, Hefei
230031, China  Science Island Branch of Graduate School, University of Science
and Technology of China, Hefei 230026, China  Ying Zhou  Institutes of
Physical Science and Information Technology, Anhui University, Hefei 230601,
China  Min Zhang  Institutes of Physical Science and Information Technology,
Anhui University, Hefei 230601, China  Liang Cao  Anhui Key Laboratory of
Condensed Matter Physics at Extreme Conditions, High Magnetic Field
Laboratory,Chinese Academy of Sciences, Hefei 230031, China  Xigang Luo
Department of Physics, University of Science and Technology of China, and Key
Laboratory of Strongly Coupled Quantum Matter Physics, Chinese Academy of
Sciences, Hefei 230026, China  Yimin Xiong111Email<EMAIL_ADDRESS>Department of Physics, School of Physics and Optoelectronics Engineering,
Anhui University, Hefei 230601, China  Hefei National Laboratory, Hefei
230028, China  Zhaorong Yang222Email<EMAIL_ADDRESS>Anhui Key Laboratory
of Condensed Matter Physics at Extreme Conditions, High Magnetic Field
Laboratory,Chinese Academy of Sciences, Hefei 230031, China  Science Island
Branch of Graduate School, University of Science and Technology of China,
Hefei 230026, China  Institutes of Physical Science and Information
Technology, Anhui University, Hefei 230601, China

###### Abstract

The ternary chalcogenide Bi2Rh3Se2 was found to be a charge density wave (CDW)
superconductor with a 2$\times$2 periodicity. The key questions regarding the
underlying mechanism of CDW state and its interplay with lattice and
electronic properties remains to be explored. Here, based on the systematic
Raman scattering investigations on single crystalline Bi2Rh3Se2, we observed
the fingerprinting feature of CDW state, a collective amplitude mode at $\sim$
39 $cm^{-1}$. The temperature evolution of Raman shift and line width for this
amplitude mode can be well described by the critical behavior of two-
dimensional (2D) Ising model, suggesting the interlayer interactions of
Bi2Rh3Se2 is negligible when CDW state is formed, as a consequence, the
quantum fluctuations play an important role at low temperature. Moreover,
temperature dependence of Raman shift for A${}_{\rm g}^{9}$ mode deviates
significantly from the expected anharmonic behavior when approaching the CDW
transition temperature $\sim$ 240 K, demonstrated that strong electron-phonon
coupling plays a key role in the formation of CDW. Our results reveal that
Bi2Rh3Se2 is an intriguing quasi-2D system to explore electronic quantum phase
transition and modulate the correlations between CDW and superconductivity.

##  I INTRODUCTION

Charge density wave(CDW) is of great interest due to its intertwined
connections with multiple quantum states of matter, such as superconductivity
(SC) and antiferromagnetism (AFM)[1, 2, 3], as well as the intriguing
mechanisms of forming ordered charge density distribution[4]. CDW usually
occurs in low-dimensional systems and involves periodic modulation of electron
density, and this electronic modulation is accompanied by lattice distortion
driven by a soft phonon mode, serving as a distinct fingerprint of the CDW
order[5]. Hence, it is of particular interest to clarify the nature of the
soft mode in CDW states: the behavior of the soft mode, which refers to the
phonon mode with a frequency approaching zero, is one of the most fundamental
and extensively studied phenomena associated with classical (thermally driven)
displacive phase transitions[6]. When a soft phonon mode condenses and causes
lattice distortions, a new Raman-active collective excitation, called
amplitude mode, emerges, offering valuable insights into the evolution and
stability of the CDW states[7, 6, 8]. Meanwhile, in low dimensional systems,
the quantum fluctuation effects are more pronounced, leading to numerous
unexpected and exotic properties with the interplay of strong correlations,
electron-phonon coupling and other quantum effects[9, 10, 11].

Recently, the compound Bi2Rh3Se2 has attracted attentions as a CDW material
with the coexistence of superconductivity(SC). Sakamoto et al. first
synthesized Bi2Rh3Se2 polycrystalline samples and revealed a CDW transition at
$T\sim$ 240 K[12]. Later Chen et al. found symmetry-forbidden reflections for
C12/m1 in selected-area electron diffraction (SAED) patterns and anomalous
pressure dependence of the phase transition temperature, suggesting a pure
structural phase transition accompanied by symmetry reduction(from C12/m1 to
P12/m1) instead of a CDW transition in Bi2Rh3Se2[13]. Recently, experiments of
Angle-resolved photoemission spectroscopy (ARPES) and infrared spectroscopy
presented new evidences of Bi2Rh3Se2 as a CDW metal with a 2 $\times$ 2
modulation of the crystal lattice[14, 15]. More recently, Wang et al.
demonstrated the existence of incommensurate and commensurate CDW orders below
250 K and 170 K in Bi2Rh3Se2[16]. Despite these experimental results, the
thermodynamic property of CDW state at low temperature, along with the
correlations to SC of Bi2Rh3Se2 have not been explored. The amplitude mode of
Raman spectra would provide an idea pathway to understand the forming
mechanism of CDW as well as its characteristics.

In this work, by Angle-Resolved polarized Raman measurement, a multitude of
CDW-induced modes were observed and studied on Bi2Rh3Se2 single crystals.
Temperature dependence of amplitude mode ($\sim$ 39 cm-1) is consistent with
the critical behavior of two-dimensional Ising model. The fast increase in the
Raman shift and line width of amplitude mode exhibit the evidences for strong
quantum fluctuations. The Raman shift of phonon-mode is clearly deviates from
the expected anharmonic model when approaching TCDW, indicating that electron-
phonon coupling plays an important role on the formation of CDW phase[17]. The
simultaneous presence of two-dimensionality, strong quantum fluctuation and
electron-phonon coupling provides the clues for understanding the coexisting
CDW and SC states in Bi2Rh3Se2.

##  II EXPERIMENT

High quality single crystals of Bi2Rh3Se2 were prepared by the method reported
in the literature[14]. Temperature-dependent resistivity was measured by using
the standard four-probe method in a Quantum Design Physical Property
Measurement System(PPMS) equipped with a 3He refrigerator. Magnetic
susceptibility from 1.8 to 300 K was measured under the field of 1 Tesla by
using a superconducting quantum interference device magnetometer(SQUID). A
freshly exfoliated single crystal with an exposed surface of ab-plane was
transferred to a JANIS-ST500 cryogenic refrigerator at a temperature of 4.2 -
300 K and a vacuum better than 1$\times$10-6 mbar for Raman measurements.
Raman scattering spectra were collected using a Renishaw inVia microscope
system with backscattering with a 532 nm laser. The laser beam is focused on
the sample with a 20 $\times$ objective, and the laser power is kept at 0.5 mW
to reduce the laser heating effect and protect the sample from damage. Single-
crystal and powder X-ray diffraction experiments were performed with a
PANalytical X′pert diffractometer using Cu K$\alpha$1 radiation ($\lambda$ =
0.15406 nm).

##  III RESULTS AND DISCUSION

Figure 1: Physical properties of Bi2Rh3Se2. The schematic crystal structure of
Bi2Rh3Se2 are shown along the b-axis (a) and c-axis (b), respectively. (c)
Temperature dependence of in-plane electrical resistivity $\rho$ of Bi2Rh3Se2,
the arrow indicates the temperature of Tc. The inset shows the derivative of
resistivity d$\rho$/dT, with dashed line marking the CDW transition
temperatures. (d) Temperature dependence of magnetic susceptibility measured
at magnetic field $\mu_{0}$H = 1 Tesla with both zero-field-cooling (ZFC) and
field-cooling (FC) process. The inset shows the derivative of susceptibility
d$\chi$/dT, with black dashed line marking the CDW transition temperatures.

Bi2Rh3Se2 has layered parkerite-type ternary chalcogenides structure stacking
along the c-axis, as displayed by the schematic figures of crystal structure
in Fig. 1(a). As shown in Fig. 1(b), the quasi-two-dimensional sheets contain
rectangular network of Rh with short interatomic Rh-Rh distances 2.86 Å along
$a$-axis and the longer one along $b$-axis[12]. At _T_ $\textgreater$
$T_{CDW}$, the Rh networks could be treated as quasi-one-dimensional (1D)
chains along the a-axis due to its shorter distance. The data of powder XRD
diffractions (at 300 K and 35 K) and analysis by standard Rietveld refinement
method are plotted in Fig. S1(a) of Supplementary Materials. The fitting
parameters are summarized in Table S1, which are consistent with the
previously reported values[12]. The single crystal XRD pattern is displayed in
Fig. S1(b). Only (00 _l_) diffraction peaks can be detected, indicating that
the exposed surface of single crystals is $ab$-plane.

Fig. 1(c) shows temperature dependence of the in-plane resistivity of a
Bi2Rh3Se2 single crystal. There is a broad hump-like resistivity anomaly below
$\sim$ 240 K, corresponding to the CDW transition reported in previous
reports[12, 14, 15, 16]. The metallic behavior of resistivity confirms the
partial gap opening at the Fermi surface in the CDW state[18]. No thermal
hysteresis can be observed in the resistivity curves between the cooling and
heating processes, indicating that this CDW phase transition is a second-order
one. The inset of Fig. 1(c) shows the derivative of resistivity d$\rho$/dT
with an extremum at 240 K. A superconducting transition at $\textit{T}_{\rm
c}$ $\sim$ 1.7 K is observed, which is higher than the reported value in
polycrystal samples[12], indicating that the SC in Bi2Rh3Se2 is strongly
affected by disorder[19, 20].

Fig. 1(d) shows the magnetic susceptibility as a function of temperature (1.8
- 300 K). A knee-like change of the slope for both field-cooling (FC) and
zero-field-cooling (ZFC) susceptibility curves appears near the CDW transition
temperature. As shown in the inset of Fig. 1(d), the derivative of
susceptibility also has an extremum at $T$ $\sim$ 240 K and no hysteresis was
observed, in agreement with the resistivity data and results of the previous
report[12]. Moreover, the susceptibility curves present a minimum value at
around 100 K. These changes of susceptibility with temperature may be related
to the change of the Fermi surface and density of states caused by the
formation of CDW states. The Hall resistivity was analysed to understand the
evolution of Fermi surfaces and density of states with temperature, as shown
in Fig. S2 of Supplementary Materials. Around TCDW $\sim$ 240 K, the carrier
density shows a slight temperature dependence, suggesting the anomalies in
resistivity and susceptibility are not caused by the change of carrier
density. At 240 K, the reconstruction of Fermi surfaces due to the formation
of CDW state may induce a partial gapping of the Fermi surface below TCDW and
orbital contribution to susceptibility, resulting in a hump-like anomaly in
resistivity and a drop in $\chi$(T). Furthermore, the carrier density
increases rapidly in commensurate CDW state and reaches a maximum value at
about 100 K, which is consistent to the feature of the susceptibility curve in
Fig. 1(d), indicating a dramatic change of Fermi surfaces and density of
states.

Figure 2: Raman spectra of Bi2Rh3Se2. (a)Schematic plot of parallel
configuration for Raman measurements. (b)Raman spectra of Bi2Rh3Se2 at T = 10
and T = 300 K, respectively. (c)Contour map of the Raman scattering intensity.
(d) Temperature dependence of Raman shift for indicated Raman modes.

In order to clarify the nature of the CDW phase transition at 240 K, a
comprehensive study of angle- and temperature-dependant Raman scattering on
Bi2Rh3Se2 single crystals was performed to probe low-energy electronic
fluctuations and phonon anomalies. Fig. 2(a) shows the schematic of the
experimental setup for the polarized Raman measurement, the angle between the
polarizations of the incident light and the principal axis (a-axis) is defined
as $\theta$. At room temperature, three acoustic and thirty nine optical
phonon modes are predicted according to DFPT calculations at $\Gamma$ point,
where 18 modes (10Ag and 8Bg) are Raman-active in the range of 0 - 300
cm-1[16]. As shown in the upper panel of Fig. 2(b), only nine Raman-active
modes were resolved (label as A${}_{\rm g}^{1}$ \- A${}_{\rm g}^{9}$, and at
300 K in our experiments, because that the intensities of other phonon modes
are too weak to be distinguished. At T = 10 K (shown in the lower panel of
Fig. 2(b)), additional six modes emerge in the Raman spectra, which are
labeled as Aamp, A${}_{\rm zf}^{1}$ \- A${}_{\rm zf}^{5}$, whose assignments
will be discussed below.

To identify the Raman vibration modes at T = 10 K, we performed Angle-Resolved
Polarized Raman measurements. The raw experimental data for all measured Raman
intensity are plotted in a contour map, as shown in Fig. S2(a). The frequency
and linewidth of all the modes were extracted by using Lorentzian or Fano
fits. When the incident light strikes along the direction of the c-axis in
parallel polarization configuration [see Fig. 2(a)], the intensity of the
collected Raman modes changes with the angle as:

$\begin{split}I\propto\lvert\vec{e}_{i}\cdot
R\cdot\vec{e}_{s}\rvert=\lvert(cos\theta\;sin\theta\;0)\left(\begin{array}[]{c}a\
b\ c\\\ d\ e\ f\\\ g\ h\ i\\\
\end{array}\right)\left(\begin{array}[]{c}cos\theta\\\ sin\theta\\\ 0\\\
\end{array}\right)\rvert^{2}\\\ =\lvert acos^{2}\theta+(b+d)sin\theta
cos\theta+esin^{2}\theta\rvert^{2},\end{split}$ (1)

where $\vec{e}_{i}$ represents the unit vector of the incident laser
polarization direction, $\vec{e}_{s}$ represents the polarization of the
scattered light direction unit vector, R is the Raman tensor. The Raman tensor
for $A_{g}$ and $B_{g}$ modes can be represented as:

$\displaystyle R_{A_{g}}=\left(\begin{array}[]{c}a\ d\ 0\\\ d\ b\ 0\\\ 0\ 0\
c\\\ \end{array}\right),R_{B_{g}}=\left(\begin{array}[]{c}0\ d\ 0\\\ d\ 0\
0\\\ 0\ 0\ 0\\\ \end{array}\right)$ (8)

Then, their Raman intensities are given by

$\displaystyle
I(A_{g})=a^{2}cos^{4}\theta+b^{2}sin^{4}\theta+4d^{2}sin^{2}\theta
cos^{2}\theta+$ (9) $\displaystyle 2adsin2\theta
cos^{2}\theta+2absin^{2}\theta cos^{2}\theta+2bdsin2\theta sin^{2}\theta,$
$I(B_{g})=0,$ (10)

From Eqs.(2) and (3), it can be concluded that Ag modes show two-fold symmetry
and Bg modes show four-fold symmetry. The angle-dependent intensity of Raman
modes can be fitted by Eq. (2) and Eq. (3), the fitting results and
experimental data of six selected modes with strong intensity are plotted in
Fig. 2(d), represented by red solid lines and black open circles. It is
notable that the new emerging modes (Aamp, A${}_{\rm zf}^{1}$ \- A${}_{\rm
zf}^{5}$) also manifest two-fold symmetry[Fig. S2(a)], implying that they are
all of A-type symmetry.

Figure. 2(d) shows the contour figure of temperature-dependent Raman intensity
for Bi2Rh3Se2. All the modes exhibit red-shift with increasing temperature due
to lattice expansion, however, the modality of the frequency shift is
different due to the different lattice anharmonicity of each mode. Among all
modes, the Aamp mode of 39.94 cm-1 (at T = 10 K) shows stronger softening and
appreciable broadening with increasing temperature than those of other modes
[See Fig. S2(b)]. Moreover, the ultrafast pump-probe study on Bi2Rh3Se2 found
a signal of collective amplitude mode at $\Omega_{A}$ = 1.25 THz (1 THz $\sim$
33.3 cm-1) at low temperature[14], which corresponds to the same phonon energy
of this Aamp mode (39.94 cm-1) observed in our work. Owing to the origin of
amplitude mode, it would disappear at the CDW transition temperature due to
the collapse of coherent CDW order[21]. But it is often heavily damped and
could not be resolved before approaching the transition temperature. The Aamp
mode at 39.94 cm-1 became broad with increasing temperature, disappearing
before 240 K. All above findings provide clear evidences that the Aamp here is
the amplitude mode of CDW phase[8].

Figure 3: Temperature dependence of amplitude mode. (a) and (b) show the
temperature dependence of Raman shift and FWHM of Aamp mode, respectively. The
solid red lines are fitting curves by using models described in the main text.
The blue and purple dashed lines are fitting curves by using mean-field(MF)
and modified mean-field model(MMF):
$\omega(T)=\omega_{0}[1-(x^{4}/3)]\sqrt{1-x^{4}}$, where $x=T/T_{CDW}$ and
TCDW = 240 K.

The temperature evolution of amplitude mode could reflect the thermodynamic
characteristics of order parameters and correlation mechanisms of the CDW
state. Recently, theoretical studies have been performed by using unbiased
quantum Monte Carlo simulations in the square lattice Hubbard-Holstein model,
to explore the competition and interplay between CDW and AFM, as well as the
coexisting SC when the electron-electron interaction is larger than a critical
value[22, 23, 24]. Besides the electron-electron interaction, a temperature
induced 2D Ising-type CDW phase transition would occur in the region of
moderate electron-phonon coupling (EPC)[25, 26].

Figure. 3 shows the Raman shift and full width at half maximum(FWHM, $\Gamma$)
of the amplitude mode as a function of temperature, respectively. The critical
exponents of the CDW phase transition can be extracted by analysing
temperature-dependent behavior of the amplitude mode[26]. A microscopic
analysis of the nature and origin of temperature-dependent mode softening can
be performed by using Rice et al.’s mean-field result[7, 27] or modified mean-
field model[28, 17] for the frequency of a CDW amplitude mode,

$\omega(T)\propto\sqrt{1-T/T_{CDW}}$ (11)

which has been successfully applied to the analysis for the behavior of CDW
soft mode in many CDW materials[29, 17], as well as the temperature dependence
of intensity of low-energy electronic bands for Bi2Rh3Se2[15]. Here, the CDW
transition temperature $T_{CDW}$ is 240 K, and we will discuss it later.

In Fig. 3 (a), the blue and purple dashed curves represent the fitting results
of mean field theory and modified mean-field model, respectively, which
dramatically deviates from our experimental data of the amplitude mode. The
simaliar deviations from the square root mean field dependence of CDW phonon
softening behavior are also observed in some two-dimensional CDW materials
materials, such as GdTe3,2H-NbSe2,1T-TiSe2,CuxTiSe2[30, 31, 32, 33], owing to
the strong CDW fluctuations[34].

Instead, the Raman shift of amplitude mode can be fitted well by using the
fitting function with a varying power $\beta$[34], as displayed by the red
line in Fig. 3(a):

$\omega(T)/\omega_{0}=(1-T/T_{CDW})^{\beta},$ (12)

where $\omega_{0}$ is the phonon frequency at $T$ $\sim$ 0 K and the fitting
parameter $\beta$ is an exponent of critical behavior, T${}_{\rm CDW}=240K$ is
extracted from the above resistivity and susceptibility data. The fit by Eq. 5
gives the values of $\omega_{0}$ = 40.03 cm-1 and $\beta$ = 0.145 $\pm$ 0.032.
The value of exponent $\beta$ matches the theoretical value of 1/8 for 2D
Ising model[35].

In Fig. 3(b), temperature dependence of the full width at half maximum(FWHM,
$\Gamma$) of the amplitude mode was also plotted and fitted by an empirical
formula[17, 33]:

$\Gamma(T)=\Gamma_{0}+A(T_{CDW}-T)^{-\gamma},$ (13)

where $\Gamma$0 is the FWHM at $T$ = 0 K, A and $\gamma$ are fitting
parameters, TCDW = 240 K. The obtained parameters are $\Gamma$0 = 4.34 cm-1, A
= 8.98 $\times$ 105 and $\gamma$ = 1.593 $\pm$ 0.244. This value of exponent
$\gamma$ is also consistent to the theoretical value $\gamma$ = 7/4 of 2D
Ising model[35].

Moreover, the intensity of amplitude mode becomes difficult to distinguish
with increasing temperature above 180 K, before it reaches CDW transition
temperature $\sim$ 240 K. At the same time, the line width of amplitude mode
broadens rapidly. The temperature dependence of the amplitude mode can be
attributed to the overdamped nature due to strong CDW fluctuations in ICCDW
state[33]. At 180 K, the frequency of the amplitude mode is about 32 cm-1,
suggesting that the amplitude mode is unlikely to be associated with the ICCDW
to CCDW transition, because in that case, the frequency should be near zero.
This overdamping of the amplitude mode make it is impossible to observe the
complete amplitude softening, but we are still able to fit the temperature-
dependent amplitude mode frequency and linewidth data, because we know the
exact value of CDW transition temperature.

According to the above temperature-dependent behavior of the amplitude mode,
we can conclude that the phase transition at $T\sim$ 240 K can be classified
into the two-dimensional Ising universality class charge transition[36],
suggesting that the interlayer interactions of Bi2Rh3Se2 are negligible for
the formation of electronic states. Meanwhile, due to the limit of
dimensionality, the effects of quantum fluctuations would be strongly
enhanced. For Ising magnetic systems, the quantum fluctuations are introduced
by applying a transverse magnetic field, creating a quantum Ising model, which
has been realized in one-dimensional systems[37, 38, 39]. For 2D systems, the
quantum Ising model can be used to describe the quantum transition[25].
However, most of the investigations is still focused on theoretical models,
more experimental researches are expected and more real systems need to be
explored. Here, in Bi2Rh3Se2, the Ising-type charge ordering is modulated by
strong quantum fluctuations, which may cause quantum transition at T = 0 K and
correspond to the low-temperature superconducting phase.

Besides the amplitude mode, with the emergence of the CDW state, in general
the zone-folding (ZF) modes would emerge in Raman response[6]. The ZF modes
correspond to a normal phonon folded into the center of the Brillouin zone due
to the establishment of the CDW phase. The electronic structure study of
Bi2Rh3Se2 by Angle-resolved photoemission spectroscopy (ARPES) experiments
show the reconstruction of electronic structure[15]. And the fact that these
modes show a minute change of frequency(as shown in [Fig. S2(b)]), more
consistent with the characteristics of a ZF mode[40, 34]. Thus, in addition to
the amplitude mode, the new-born modes below 150 K could be labelled as ZF
modes(A${}_{\rm zf}^{1}$-A${}_{\rm zf}^{5}$).

Figure 4: Temperature-dependent A${}_{\rm g}^{7}$ and zone-fold mode A${}_{\rm
zf}^{5}$.(a)-(c) Show the temperature dependence of Raman shift and FWHM of
A${}_{\rm g}^{7}$ and A${}_{\rm zf}^{5}$ modes, respectively. The solid lines
are fitting curves by using anharmonic phonon model. (d) Temperature
dependence of lattice parameters a, b, and c, the solid line is a guide to the
eye.

In Raman spectra, isotopic and/or anharmonic effects, and electron-phonon
interactions cause the change of linewidth. Temperature dependance of the
phonon mode energy ($\omega$) and linewidth ($\Gamma$) are usually governed by
phonon-phonon interaction (anharmonic effects)[41, 42, 43, 44]. In order to
further quantitatively evaluate the lattice dynamic properties of Bi2Rh3Se2,
we fitted the temperature dependence of $\omega$ and $\Gamma$ by using the
following anharmonic phonon model[43]:

$\omega(T)=\omega_{0}-B(1+\frac{2}{e^{\frac{\hbar\omega_{0}}{2k_{B}T}}-1}),$
(14)
$\Gamma(T)=\Gamma_{0}+C(1+\frac{2}{e^{\frac{\hbar\omega_{0}}{2k_{B}T}}-1}),$
(15)

where B and C are anharmonic constant and kB is the Boltzmann constant. In
some CDW materials, the phonons may also be coupling with other fundamental
excited states(i.e. electrons)[43, 45], consequently an additional term
$\Gamma_{0}$ must be included.

Two most intense modes (A${}_{\rm g}^{7}$ and A${}_{\rm zf}^{5}$) were
selected to demonstrate the evolution of Raman modes with temperature below
the CDW transition in Bi2Rh3Se2. The corresponding fitting parameters of all
the modes are given in Table S2. As shown in Fig. 4, both A${}_{\rm g}^{7}$
and A${}_{\rm zf}^{5}$ modes can be well fitted by the anharmonic phonon
model, suggesting a stable CDW state with the robust lattice distortions below
TCDW[17]. A${}_{\rm zf}^{5}$ mode fades away well below the CDW transition
temperature due to the strong thermal fluctuations. It is worth noting that
the A${}_{\rm g}^{7}$ mode deviates from the fitting curve when it is close to
the CDW transition temperature, implying the temperature-induced anharmonicity
effect cannot describe the change of slope at TCDW. And the A${}_{\rm g}^{7}$
exhibits blue shift with the increasing temperature, as shown in Fig. 4(a), by
which the effect of lattice expansion can be ruled out. Owing to the above
temperature dependence, we can conclude that the dramatic change in A${}_{\rm
g}^{7}$ mode at 240 K may come from strong temperature-dependent electron-
phonon interactions[43].

Electron-phonon coupling is known to lead to asymmetric peak shapes, which is
attributed to Breit-Wigner-Fano (BWF) resonance (or Fano resonance). As shown
in Fig. S2(c), the peak of A${}_{\rm g}^{7}$ mode shows a slight asymmetric,
which can be fitted well by the Fano model [46]:

$I=I_{0}\frac{[1+2(\omega-\omega_{0})/q\Gamma]^{2}}{1+[2(\omega-\omega_{0})/\Gamma]^{2}},$
(16)

where I0 is the peak intensity at the peak position $\omega_{0}$, $\Gamma$
corresponds to the full width at half maximum (FWHM). 1/$\mid$q$\mid$ is the
degree of electron-phonon coupling which describes the departure of the line
shape from a symmetric Lorentzian function. The positive and negative signs of
q indicate the relative position of the asymmetry, respectively. The
Lorentzian shape is obtained when 1/$\mid$q$\mid$$\rightarrow$0\. The
1/$\mid$q$\mid$ values before and after the phase transition are almost the
same as displayed by the dark yellow line in Fig. 4(b). This means that the
main contribution to the temperature dependence of the linewidth comes from
the phonon-phonon(anharmonic) interactions, because the electron-phonon
contribution is temperature independent(1/$\mid$q$\mid$ $\sim$ const)[43, 16,
47]. At $T\sim$ 240 K, 1/$\mid$q$\mid$ reaches the largest value, which
further indicates that this phase transition is related to the electron-phonon
coupling. From ARPES measurements by Liu et al.[15], a peak-dip-hump feature
can be found near the Fermi momentum, which is manifested the existence of
electron-phonon coupling. Therefore, by the analysis of Raman spectra, we
believe that electron-phonon coupling may play an important role in the
formation of CDW phase in Bi2Rh3Se2. Beside the electron-phonon coupling, the
change of electronic structure could be a possible contribution to the change
of lineshape and linewidth of Raman spectra. According to the previous
literature of ARPES experiments, the change of electronic structure was
observed below the CDW transition temperature $\sim$ 240 K[14]. However, the
change of density of states may occur at a lower temperature due to the
gradually enhanced hybridization between replica bands and primary bands with
the decreasing temperature. The Hall effect is able to provide simple
indications about the electronic structure and the density of states at Fermi
level. Hall resistivities at various temperatures were measured and presented
in Fig. S2 of Supplementary Material. It shows linear magnetic field
dependence when $T\geq$ 30 K, indicating the electrical transport is dominated
by the electron carriers, although the ARPES results reveal both electron and
hole pockets at the Fermi level. Meanwhile, a non-linear dependence of Hall
resistivity was observed at low temperatures, indicating a change of Fermi
surfaces and transport properties. The low temperature Hall conductivities
were fitted by two-carrier model with formula Eq. Sx, as shown in Fig. Sx. The
carrier density at different temperatures was extracted and plotted in Fig.
Sx. It exhibits a slight temperature dependence between 200 and 300 K, due to
the thermal effect of lattice with changing temperature. However, there is no
clear change at 240 K, the CDW transition temperature, implying that the
transition is dominated by the electron-phonon coupling, and the contribution
of electronic structure could be negligible. In fact, the reasons influencing
the Fano line shape also include the density of states and the light-electron
matrix element. Therefore, further efforts will be required in the future to
eliminate their effects.

To understand the evolution of lattice structure around the CDW transition,
the powder XRD at variation temperature from 35 K to 300 K were measured and
shown in Figs. S3(a). The derived lattice parameters and volume as functions
of temperature are presented in Fig. 4(d) and Fig. S3(b). No structural
transition was observed with changing temperature. A distinct kink was
observed at $\sim$ 240 K, corresponding to the CDW phase transition. This
indicates the correlation between lattice modulation and CDW instability.

##  IV CONCLUSIONS

To summary, by using temperature-dependent polarized Raman spectra, a second-
order CDW phase transition with distinctive features near TCDW = 240 K has
been explored on single crystals of Bi2Rh3Se2. One CDW amplitude mode and five
ZF modes are identified in terms of the present Raman scattering
configurations, which could be fingerprinting characteristics for the CDW
state. Temperature-dependent behavior of the amplitude mode indicates that the
CDW state below $T\sim$ 240 K can be classified to two-dimensional Ising
universality model, where the interlayer interactions could be negligible for
the formation of electronic states of Bi2Rh3Se2. Meanwhile, the simultaneous
strong quantum fluctuations make Bi2Rh3Se2 an ideal 2D system to study
electronic quantum transition and the effects of quantum fluctuations on
charge ordering. Furthermore, the unexpected phonon shift and coupling
constant around CDW transition temperature clearly demonstrate the strong
electron-phonon coupling may be the driving force of the transition.
Consequently, Bi2Rh3Se2 is a compound with multiple exotic properties, for
instant CDW, SC, strong quantum fluctuations and electron-phonon coupling,
which deserves further investigations.

###### Acknowledgements.

This work was financially supported by the National Key Research and
Development Program of China (Grant No.2022YFA1602603,2021YFA1600200), the
National Natural Science Foundation of China (NSFC) (Grants No. 12374049, No.
12204420, No. 12204004, No. 12174395, No. U19A2093 and No. 12004004), the
Natural Science Foundation of Anhui Province (No. 2308085MA16, No.
2308085QA18), the Innovation Program for Quantum Science and Technology (No.
2021ZD0302802). A portion of this work was supported by the High Magnetic
Field Laboratory of Anhui Province under Contract No. AHHM-FX-2021-03.This
work was supported by Systematic Fundamental Research Program Leveraging Major
Scientific and Technological Infrastructure, Chinese Academy of Sciences under
contract No. JZHKYPT-2021-08.Yonghui Zhou was supported by the Youth
Innovation Promotion Association CAS (Grant No. 2020443), Y. X. is supported
by Anhui University through the start-up project (Project No. S020318001/020).

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