#  Lattice dynamics of the Ta2Ni(Se1-xSx)5 excitonic insulator

Mai Ye<EMAIL_ADDRESS>Department of Physics and Astronomy, Rutgers
University, Piscataway, NJ 08854, USA  Pavel A. Volkov
<EMAIL_ADDRESS>Department of Physics and Astronomy, Rutgers
University, Piscataway, NJ 08854, USA  Himanshu Lohani  Department of Physics,
Technion - Israel Institute of Technology, Haifa 32000, Israel  Irena Feldman
Department of Physics, Technion - Israel Institute of Technology, Haifa 32000,
Israel  Minsung Kim  Department of Physics and Astronomy, Rutgers University,
Piscataway, NJ 08854, USA  Amit Kanigel  Department of Physics, Technion -
Israel Institute of Technology, Haifa 32000, Israel  Girsh Blumberg
<EMAIL_ADDRESS>Department of Physics and Astronomy, Rutgers
University, Piscataway, NJ 08854, USA  National Institute of Chemical Physics
and Biophysics, 12618 Tallinn, Estonia

###### Abstract

Recently, we employed electronic polarization-resolved Raman spectroscopy to
reveal the strongly correlated excitonic insulator (EI) nature of Ta2NiSe5
[Volkov et al., npj Quant. Mater. 6, 52 (2021)], and also showed that for
Ta2Ni(Se1-xSx)5 alloys the critical excitonic fluctuations diminish with
sulfur concentration $x$ exposing a cooperating lattice instability that takes
over for large $x$ [Volkov et al., arXiv:2104.07032]. Here we focus on the
lattice dynamics of the EI family Ta2Ni(Se1-xSx)5 ($x=0$, …, 1). We identify
all Raman-active optical phonons of $A_{g}$ (fully symmetric) and $B_{2g}$
($ac$-quadrupole-like) symmetries ($D_{2h}$ point group) and study their
evolution with temperature and sulfur concentration. We demonstrate the change
of selection rules at temperatures below the orthorhombic-to-monoclinic
transition at $T_{c}(x)$ that is related to the EI phase. We find that
$T_{c}(x)$ decrease monotonically from 328 K for Ta2NiSe5 to 120 K for Ta2NiS5
and that the magnitude of lattice distortion also decreases with the sulfur
concentration $x$. For $x<0.7$, the two lowest-frequency B2g phonon modes show
strongly asymmetric lineshapes at high temperatures due to Fano interference
with the broad excitonic continuum present in a semimetallic state. Within the
framework of extended Fano model, we develop a quantitative description of the
interacting exciton-phonon excitation lineshape, enabling us to derive the
intrinsic phonon parameters and determine the exciton-phonon interaction
strength, that affects the transition temperature $T_{c}(x)$ . While at low
temperatures the intrinsic phonon parameters are in good agreement with the ab
initio calculations and the anharmonic decay model, their temperature
dependencies show several anomalous behaviors: (i) Frequencies of B2g phonons
harden pronouncedly upon cooling in vicinity of $T_{c}(x)$ for $x<0.7$
semimetals and, in contrast, soften monotonically for Ta2NiS5 semiconductor;
(ii) The lifetime of certain phonons increases strongly below $T_{c}(x)$ for
$x<0.7$ revealing the gap opening in the broken symmetry phase; (iii) For most
modes, the intensity shows rather strong temperature dependence hat we relate
to the interplay between electronic and phononic degrees of freedom. For
Ta2NiSe5 we also observe signatures of the acoustic mode scattered assisted by
the structural domain walls formed below $T_{c}$. Based on our results, we
additionally present a consistent interpretation of the origin of oscillations
observed in time-resolved pump-probe experiments.

††preprint: This is a preprint of an article published in the Physical Review
B with the Editor’s Suggestion.††preprint: The final authenticated version is
available online at 10.1103/PhysRevB.104.045102.

##  I Introduction

In a narrow-gap semiconductor or semimetal, if the exciton binding energy
exceeds the band gap, the Coulomb attraction between electrons and holes
favors spontaneous formation of a macroscopic number of excitons. These
bosonic quasiparticles then form a coherent state, opening an interaction-
induced gap and leading to a insulating phase, named excitonic insulator [1,
2, 3]. If the electron and hole states belong to bands of different symmetry,
the resulting excitonic insulator state is expected to break the crystal
lattice symmetry.

Consider the case of a semimetal. If the band gap is indirect, i.e. the
valence band maximum and the conduction band minimum are located at different
places in the $k$-space, condensation of excitons leads to a charge-density-
wave (CDW) state and accompanying crystal distortion which breaks
translational symmetry [3]. The CDW wavevector is then given by the spanning
vector that connects the valence band maximum to the conduction band minimum.
One such example is semimetal 1T-TiSe2, which has been proposed to hold an
excitonic insulating state below 190 K [4, 5, 6, 7]. In contrast, if the band
gap is direct, excitonic condensation only results in lattice distortion which
breaks point-group symmetry. In this situation, the size of the unit cell does
not change but its shape changes. Ta2NiSe5 is a candidate for excitonic
insulator of this kind.

Ta2NiSe5 shows a second-order structural phase transition at $T_{c}=326$ K [8,
9]. In the high temperature phase, this material has a orthorhombic structure
(point group $D_{2h}$); in the low-temperature phase, the structural is
monoclinic (point group $C_{2h}$) with the angle $\beta$ between $a$ and $c$
directions deviating from 90∘, Fig. 1. Below $T_{c}$ , the valence-band top
flattens as discovered by angle-resolved photoemission spectroscopy (ARPES)
studies [10, 11, 12]. Band flatness at low temperature is viewed as a
characteristic feature of the excitonic insulator ground state. Moreover,
studies of nonequilibrium dynamics have demonstrated photoinduced enhancement
of excitonic order [13, 14], photoinduced multistage phase transitions [15],
ultrafast reversal of excitonic order [16], and coherent order-parameter
oscillations [17].

Figure 1:  Crystal structure of Ta2NiSe5. The frame in (a) indicates the unit
cell.

The lattice degrees of freedom play an important role in the physics of
Ta2NiSe5. In particular, the orthorhombic-to-monoclinic structural change can
be induced via electron-phonon coupling [18] even if the origin of the
transition is excitonic [19]. Moreover, a recent study combining ultrafast
experiments with calculations proposes that the phase transition in Ta2NiSe5
is structural in its nature [20]. Thus, understanding the underlying phsyics
of excitonic insulator candidates requires the lattice dynamics to be examined
in a detailed way. Interestingly, substitution of Se with S results in a whole
family of compounds Ta2Ni(Se1-xSx)5, where the orthorhombic-to-monoclinic
transition has been suggested to be suppressed with $x$ [8]. While it is
known, that S substitution leads to an enhanced direct band gap [21], which is
detrimental to the EI state, its effects on the lattice modes has not been
systematically studied.

Because the inversion symmetry of Ta2Ni(Se1-xSx)5 is preserved across $T_{c}$
, the soft mode of this zero-wavevector structural transition, if any, is
Raman active. Polarization resolved low frequency Raman spectroscopy,
therefore, is particularly suitable for studying the physics of the phase
transition, and for evaluating the contribution of the lattice vibrations to
the transition. This experimental method offers both the high energy
resolution and the ability to selectively probe bosonic excitations in
different symmetry channels [22].

In this work, we present a systematic study of the lattice dynamics for the
family of Ta2Ni(Se1-xSx)5 ($x=0$, 0.25, 0.67, and 1) pristine crystals and
alloys to quantitatively explore the spectral parameters of the phonons. We
find the change of selection rules, indicating an orthorhombic-to-monoclinic
structural transition at $T_{c}(x)$, which is finite for all $x$. The
transition temperature $T_{c}(x)$ and the magnitude of lattice distortion both
decrease with sulfur concentration $x$.

For Ta2Ni(Se1-xSx)5 with $x\leq 0.67$, the two lowest-energy symmetry-breaking
B2g phonon modes exhibit strong asymmetric lineshape above the transition
temperature, resulting from the coupling between these modes and an excitonic
continuum of the same symmetry. We develop an extended model to analyze the
physics of this Fano-type interference effect. Using this model, we
disentangle the excitonic and phononic contributions to the Raman response,
and illustrate the effect of the coupling between them: apart from
renormalizing the bare phononic and excitonic responses, this effect also
results in an additional interference term in the total Raman response. We
find that the B2g phonon modes exhibit no intrinsic softening on cooling
towards the transition temperature. In contrast, the excitonic response shows
clear soft-mode behavior [23, 24]. Additionally, many modes exhibit an
anomalous increase in linewidth above $T_{c}$, suggesting the closing of the
EI gap. For Ta2NiSe5 below $T_{c}$ , we also observe signatures of the
acoustic excitations with finite momenta, enabled by the formation of a quasi-
periodic structure of domain walls.

For Ta2NiS5 , on the contrary, the phonon modes in the symmetry-breaking
channel maintain symmetric lineshape, but their frequency anomalously
decreases upon cooling. Because the structural change for Ta2NiS5 is very
weak, no anomaly of phonon frequency or width is detected around its
structural transition temperature.

For all compositions, we find that most phonons exhibit strong dependence of
their intensity on temperature, and present arguments in favor of this being a
result of a coupling to electronic excitations.

Additionally, we perform density functional theory (DFT) calculations of the
lattice dynamics for Ta2NiSe5 and Ta2NiS5 . The phonon frequencies agree well
with the experiment, and the obtained displacement patterns of the optical
modes allow to explain the differences of the couplings of the three non-
symmetric phonons to the excitonic continuum.

Finally, we demonstrate that our results are consistent with the modes
observed in time-resolved experiments. We show however, that the signatures of
the Fano-shaped phonons and the excitonic continuum above $T_{c}$ may be hard
to observe in that type of experiments.

The rest of this paper is organized as follows. In Sec. II we describe the
sample preparation and experimental setup. In Sec. III we show an overview of
the phonon spectra of Ta2Ni(Se1-xSx)5 family, and compare the measured phonon
frequencies at low temperatures with the calculated values. In Sec. IV from
the temperature dependence of the phonon intensity we obtain phase transition
temperatures, which are compared with the resistance measurements. In Sec. V
we analyze the Fano interference feature of B2g-symmetry Raman response. In
Sec. VI we present the temperature dependence of phonon-mode parameters: the
results for B2g-symmetry modes are given in SubSec. VI.1, and those for Ag-
symmetry modes are given in SubSec. VI.2. In Sec. VII we convert Raman
response from frequency domain to time domain, and discuss relevant time-
resolved studies on Ta2NiSe5. In Sec. VIII we provide a summary of the
observations and their implications. Further discussion is provided in
appendices: the methods used for the estimation of laser heating rate in
Appendix. A; the mathematical formalism for the generalized Fano model in
Appendix B, and more illustration of this model in Appendix C; the fitting
results of the anharmonic decay model for the phonon modes in Appendix D.

##  II Experimental

Single crystals of Ta2Ni(Se1-xSx)5 family were grown by chemical vapor
transport method. Elemental powders of tantalum, nickel, selenium and sulfur
were mixed with stoichiometric ratio and then sealed in an evacuated quartz
ampule with a small amount of iodine as the transport agent. The mixture was
placed in the hot end of the ampule ($\sim$950∘C) under a temperature gradient
of about 10∘C/cm. After about a week mm-sized needle-like single crystals were
found at the cold end of the ampule. The crystals are shiny and cleave easily.
We used x-ray diffraction and electron dispersive X-ray spectroscopy to verify
the exact composition of the crystals and their uniformity. The samples for
resistance and Raman measurements are from the same batch.

The resistance is measured along a axis in a four-probe configuration using a
Quantum Design PPMS system.

The TEM images of domain structures are taken on a FEI Titan Themis G2 system
using a Gatan double-tilt cryo-stage.

For Raman measurements, the samples were cleaved in ambient conditions to
expose the $ac$ plane; the cleaved surfaces were then examined under a
Nomarski microscope to find a strain-free area. Raman-scattering measurements
were performed in a quasi-back-scattering geometry from the samples mounted in
a continuous helium-gas-flow cryostat.

For acquisition of the low frequency Raman response required for this study,
we used a custom fast f/4 high resolution 500/500/660 mm focal lengths triple-
grating spectrometer (with 1800 mm-1 master holographic gratings comprised of
(i) aberration corrected subtractive stage providing a reliable 12 orders-of-
magnitude stray light rejection at as low frequency as 4 cm-1 from the elastic
line, (ii) a third stage monochromator, and (iii) a liquid-nitrogen-cooled
charge-coupled device (CCD) detector (Princeton Instruments). All the acquired
data were corrected for the spectral response of the spectrometer. Three slit
configurations were used: 100 $\mu$m slit width providing 0.19 meV spectral
resolution; 50 $\mu$m slit width rendering 0.10 meV spectral resolution; and,
for high resolution data, 25 $\mu$m slit width rendering 0.06 meV spectral
resolution.

For polarization optics, a Glan-Taylor polarizing prism (Melles Griot) with a
better than 10-5 extinction ratio to clean the laser excitation beam and a
broad-band 50 mm polarizing cube (Karl Lambrecht Corporation) with an
extinction ratio better than 1:500 for the analyzer was used. Two polarization
configurations were employed to probe excitations in different symmetry
channels. The relationship between the scattering geometries and the symmetry
channels [22] is given in Table 1.

Table 1: The Raman selection rules in the high-temperature orthorhombic (point group D2h) and low-temperature monoclinic (point group C2h) phases. Upon the reduction of symmetry from D2h to C2h, the Ag and B2g irreducible representations of D2h group merge into the Ag irreducible representation of C2h group. Scattering | Symmetry Channel | Symmetry Channel
---|---|---
Geometry | (D2h group) | (C2h group)
aa | Ag | Ag
ac | B2g | Ag

The 647 nm line from a Kr+ ion laser was used for excitation. Incident light
was focused to an elongated along the slit direction 50$\times$100 $\mu$m2
spot. For data taken below 310 K, laser power of 8 mW was used. To reach
temperature above 310 K, we kept the environmental temperature at 295 K and
increased laser power to reach higher sample temperature in the excitation
spot. All reported data were corrected for laser heating in two mutually
consistent ways [Appendix. A]: (i) by Stokes/anti-Stokes intensity ratio
analysis, based on the principle of detailed balance, (ii) by checking laser
power that is inducing the phase transition. In addition to that, we performed
a thermoconductivity model calculation that suggests a linear scaling of the
heating rate with the beam spot size.

The first-principle calculations were performed using DFT within Projector
Augmented-Wave (PAW) formalism [25] and Perdew-Burke-Ernzerhof (PBE)
parametrized exchange-correlation energy functional [26]. For Ta2NiSe5, we
used the implementation of the Quantum ESPRESSO package [27], using the
pseudopotentials generated by Dal Corso [28] and with grimme-d3 Van-der-Waals
correction [29] included. For Ta2NiS5, the implementation of the Vienna Ab
initio Simulation Package (VASP) code [30, 31] was used. The phonon
frequencies for both cases were calculated by finite displacement method as
implemented in PHONOPY [32]. The initial structures subject to relaxation for
Ta2NiSe5 and Ta2NiS5 have been taken from experiments on the high-temperature
orthorhombic phase, Ref.[33] and Ref.[34], respectively.

##  III Overview

In this section we show an overview of the phonon spectra of Ta2Ni(Se1-xSx)5
family, and compare the measured phonon frequencies at low temperatures with
the calculated values.

The phonon spectra of Ta2Ni(Se1-xSx)5 family ($x$=0, 0.25, 0.67, and 1) are
summarized in Fig. 2. For stoichometric compositions at high temperature, we
observe $8$ modes in $aa$ geometry and $3$ modes in $ac$ geometry, in
agreement with the 8 Ag and 3 B2g Raman-active phonon modes expected in the
high-temperature orthorhombic phase (space group $Cmcm$): the Ag modes appear
in $aa$ scattering geometry and the B2g modes in $ac$ scattering geometry
[Table. 1]. Except for the Ta2NiS5 sample, the two lowest-energy B2g modes,
B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$, exhibit strongly broadened and
asymmetric lineshapes at high temperature. This anomalous broadening is almost
absent for Ta2NiS5 (note the logarithmic intensity scale in Fig. 2).

Figure 2:  Raman response $\chi^{\prime\prime}$ in the $aa$ and $ac$
scattering geometries for Ta2Ni(Se1-xSx)5 family, plotted in semi-log scale.
The phonon modes are classified by the Ag and B2g irreducible representations
of $D_{2h}$ point group. The Ag and B2g modes of Ta2NiSe5 are labeled by red
and green colors, respectively; the Ag and B2g modes of Ta2NiS5 are labeled by
purple and blue colors, respectively. For the samples with $x$ = 0.25 and
0.67, the modes are labeled by the color scheme of Ta2NiSe5 if its energy is
close to that of Ta2NiSe5 modes, or the color scheme of Ta2NiS5 if its energy
is close to that of Ta2NiS5 modes. The spectral resolution is 0.19 meV for the
$ac$ spectra at high temperature, and 0.06 meV for the $ac$ spectra at 35 K
and all $aa$ spectra.

At low temperature, a transition into a monoclinic phase (space group $C2/c$)
occurs. In the low-temperature phase, because of the mirror symmetry breaking,
all Raman-active phonon modes appear in both $aa$ and $ac$ scattering
geometries [Table. 1]. All of the expected Raman-active modes were observed:
they are identified by dashed lines in Fig. 2; the additional weak spectral
features result from the second-order scattering. For example, the weak
feature at 34.1 meV in $aa$ scattering geometry for Ta2NiSe5 corresponds to a
second-order scattering feature of the 16.8 meV B${}_{2g}^{(2)}$ mode.

The measured phonon frequencies for Ta2NiSe5 and Ta2NiS5 at 35 K generally
match well with the calculated values [Table. 2]. As the calculations were
carried out in the orthorhombic phase, this suggests that monoclinicity by
itself does not strongly affect the phonon frequencies. Larger than 10%
discrepancies between the measured and calculated values appear only for the
B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$ modes of Ta2NiSe5 and the
B${}_{2g}^{(1)}$ mode of Ta2NiS5. As discussed in Sec. V, these modes exhibit
an anomalous behavior that can be attributed to their strong coupling to the
excitons.

Table 2:  Comparison of the experimentally measured phonon energies at 35 K and the calculated values for Ta2NiSe5 and Ta2NiS5. Unit is meV; modes are identified by their irreducible representations in the orthorhombic phase, where 8 Ag modes and 3 B2g modes are expected. | Ta2NiSe5 | Ta2NiS5
---|---|---
Mode | Exp. | Calc. | Exp. | Calc.
A${}_{g}^{(1)}$ | 4.22 | 4.22 | 5.01 | 4.82
B${}_{2g}^{(1)}$ | 8.66 | 7.61 | 6.50 | 7.66
A${}_{g}^{(2)}$ | 12.28 | 11.87 | 16.13 | 15.18
B${}_{2g}^{(2)}$ | 15.25 | 10.14 | 16.41 | 16.20
A${}_{g}^{(3)}$ | 16.76 | 15.53 | 19.13 | 18.27
B${}_{2g}^{(3)}$ | 18.38 | 19.39 | 32.96 | 33.23
A${}_{g}^{(4)}$ | 22.17 | 21.91 | 33.82 | 33.82
A${}_{g}^{(5)}$ | 24.14 | 24.09 | 36.50 | 36.36
A${}_{g}^{(6)}$ | 27.02 | 26.34 | 40.07 | 40.13
A${}_{g}^{(7)}$ | 29.39 | 29.01 | 43.10 | 43.07
A${}_{g}^{(8)}$ | 36.32 | 37.47 | 49.49 | 49.62

For the alloy compositions ($x=0.25$ and 0.67), the phonon modes show larger
linewidth compared with that for the stoichometric compositions. Moreover, the
B${}_{2g}^{(3)}$ and A${}_{g}^{(3)}$-A${}_{g}^{(8)}$ modes exhibit two
distinct frequencies: one close to the frequency of that mode in Ta2NiSe5, and
the other close to the frequency in Ta2NiS5. Such behavior is commonly
observed for alloys in which the frequencies of the same phonon mode in the
two end-point materials differ substantially [35, 36, 37]. The doping
dependence of the phonon frequencies measured at 35 K is presented in Fig. 3.
The frequencies of the phonon modes for Ta2NiSe5 are consistent with the
recent experimental studies [38, 39, 40, 41, 42] and the calculated values
[43].

Figure 3:  The doping dependence of the Raman-active phonon frequencies at 35
K for Ta2Ni(Se1-xSx)5 family. For the doped samples, the B${}_{2g}^{(3)}$ and
A${}_{g}^{(3)}$-A${}_{g}^{(8)}$ phonons exhibit two-mode behavior (i.e. two
peaks in the spectrum), characteristic of alloys [35, 36, 37].

##  IV Phase transition

In this section we deduce the phase transition temperatures from the observed
change in selection rules in Raman response and compare the results with the
resistance measurements.

Below $T_{c}$, the 11 phonon modes are allowed by symmetry to appear in both
scattering geometries [Table.1]; thus, 8 additional modes appear in $ac$
geometry and 3 in $aa$. We call the appearance of phonon modes below $T_{c}$
in the orthogonal to allowed above $T_{c}$ scattering geometry as phonon-mode
”leakage”. As shown in Fig. 2, all four samples exhibit phonon-mode ”leakage”
at low temperature, indicating presence of a point group symmetry breaking
structural phase transition.

In Fig. 4 we show the temperature dependence of the ”leakage” of
A${}_{g}^{(1)}$ and B${}_{2g}^{(1)}$ modes for Ta2Ni(Se1-xSx)5 family. The
”leakage” of other modes is given in Sec. VI. For each sample, the ”leakage”
of different modes appear below the same temperature.

Figure 4:  Temperature dependence of the integrated intensity of
A${}_{g}^{(1)}$ and B${}_{2g}^{(1)}$ modes in the ’forbidden’ scattering
geometry ($ac$ and $aa$, respectively) for (a) Ta2NiSe5, (b) Ta2NiS5, (c)
Ta2Ni(Se0.75S0.25)5, and (d) Ta2Ni(Se0.33S0.67)5.

We assign the temperature of the ”leakage” onset as the structural phase
transition temperature $T_{c}$ . The phase transition temperatures for
Ta2Ni(Se1-xSx)5 family are given in Table 3. The transition temperatures
$T_{c}(x)$ decrease with sulfur doping $x$. Moreover, for larger sulfur
content $x$ the phonon intensities in the ’forbidden’ scattering geometry also
decrease. This quantity is proportional to the square of the order parameter
(OP), which can be represented by, e.g., the deviation of the angle $\beta$
between $a$ and $c$ axes [Fig. 1 (b)] from 90∘. While the general monotonic
growth of the leakage intensity on cooling is consistent with these
expectations, the details of temperature dependence can vary for different
modes [Fig. 4(a)]. The decreasing trend of the phonon intensity in the
’forbidden’ scattering geometry with sulfur content $x$ therefore indicates
that the structural distortion becomes weaker with higher sulfur
concentration.

Table 3: The structural phase transition temperature of Ta2Ni(Se1-xSx)5 family determined by observing phonon-mode ”leakage” in Raman spectra. Sulfur Content $x$ | Transition Temp. (K)
---|---
0 | 328$\pm$5
0.25 | 260$\pm$10
0.67 | 170$\pm$10
1 | 120$\pm$10

A complementary way to determine the phase transition temperature was
demonstrated in [8], where anomalies in the temperature dependence of
resistance have been detected at $T_{c}(x)$. In Fig. 5 we show the resistance
data analysis for the samples from the same batch as used for the Raman
measurements. For Ta2NiSe5 and Ta2Ni(Se0.75S0.25)5, the temperature at which
the temperature derivative of resistance displays a kink and that of transport
activation gap shows a peak, coincides with the Raman-determined transition
temperature. For Ta2Ni(Se0.33S0.67)5 and Ta2NiS5, though, the features in
resistance data are less pronounced. The latter can be anticipated from their
much more insulating character, potentially masking a small change of
resistance due to transition on top of a large background resistance.

Figure 5:  Temperature dependence of the resistance along $a$ axis for
Ta2Ni(Se1-xSx)5 family. (a) The resistance, plotted in semi-log scale. (b) The
temperature derivative of resistance, plotted in semi-log scale. (c) The
temperature derivative of the transport activation gap
$E_{g}=k_{B}T\log_{10}{[R(T)/R(360K)]}$. The transition temperatures
determined from Raman measurements are marked by dashed lines for comparison.

##  V Fano Interference in the B2g symmetry channel

In this section we analyze the asymmetric lineshape of the B2g-symmetry phonon
modes observed for $x<0.7$ in the high temperature phase. We describe the
asymmetric lineshape and its temperature dependence in SubSec. V.1. To fit the
spectra, we develop a generalized Fano model, discussed in SubSec. V.2, in
which the asymmetric broadening results from the coupling of the phonon modes
to an excitonic continuum. Besides the optical phonon modes, the excitonic
continuum also couples to the acoustic phonon mode. We discuss how such
coupling enhances the transition temperature $T_{c}(x)$ , and leads to
softening of the B2g -symmetry acoustic mode in SubSec. V.3. The appearance of
a quasi-periodic domain wall structure below $T_{c}$ introduces additional
complexity, and in SubSec. V.4, we discuss how the fitting model must be
modified to account for low-energy spectral features which are absent above
$T_{c}$ . The fitting results for Ta2NiSe5, relevant to the excitonic
continuum, are presented and interpreted in SubSec. V.5 (those related to the
intrinsic phonon properties are shown later in Sec. VI). It is demonstrated
that the strengths of the interactions between individual phonons and
excitonic continuum are consistent with the displacement patterns deduced from
ab initio calculations. For comparison, the fitting results for the alloy
compositions are given in SubSec. V.6.

###  V.1 Temperature dependence of asymmetric Fano lineshapes

The temperature dependence of the spectra measured in $ac$ scattering geometry
for Ta2NiSe5 is shown in Fig. 6. Above $T_{c}$ , the B${}_{2g}^{(1)}$ and
B${}_{2g}^{(2)}$ modes have noticeably asymmetric lineshapes, while the
lineshape of the B${}_{2g}^{(3)}$ mode remains quite symmetric. Upon
approaching $T_{c}$ on cooling, the spectral weight of low-frequency continuum
displays an enhancement, and the apparent linewidths of the B${}_{2g}^{(1)}$
and B${}_{2g}^{(2)}$ modes increase. Below $T_{c}$ , the modes recover the
conventional Lorentzian lineshapes.

Figure 6:  Temperature dependence of Raman response $\chi^{\prime\prime}$ in
the $ac$ scattering geometry for Ta2NiSe5. The data is shown by black dots,
along with the Fano fits [Eq. (35)] shown as red curves. The spectral
resolution is 0.19 meV for panels (a-e), 0.10 meV for panels (f-g), and 0.06
meV for panel (h). The Fano model describes the coupled dynamics of the
B${}_{2g}^{(1)}$-B${}_{2g}^{(3)}$ phonon modes and the overdamped excitonic
continuum. The bare phonon modes [Eq. (1)] are shown by green curves, and the
bare excitonic continuum [Eq. (2)] is shown by blue curves. Below $T_{c}$ ,
especially at 300 K, additional spectral weight is observed at low energies,
resulting from the emergence of finite-energy acoustic modes due to scattering
off structural domain walls (see Fig.9). The purple curves show the coupled
response of the acoustic modes and the excitonic continuum. As the excitonic
continuum is suppressed on cooling below $T_{c}$ , the lineshapes of the
phonons become essentially symmetric.

An asymmetric lineshape arises due to Fano interference when a sharp phonon
interacts with a broad continuum. In this case, the continuum is due to the
overdamped exciton fluctuations [23, 24], and the exciton-phonon interaction
is caused by modulation of electronic bands near the Fermi surface by the
B2g-symmetry lattice vibrations.

###  V.2 The generalized Fano model for data above $T_{c}$

We describe the Fano model that includes the coupling between the
B${}_{2g}^{(1-3)}$ phonon modes and the continuum; more details can be found
in Appendix B.

We first discuss the Raman response of the phonons. The Raman scattering
intensity is related to the fluctuation spectrum of the phonon coordinate
$\langle Q_{\omega}Q_{-\omega}\rangle$ [44], which is given by the imaginary
part of the phonon coordinate susceptibility $\chi^{\prime\prime}_{p}(\omega)$
times the Bose factor. The phononic Raman response function consequently has
the form:

$\chi^{\prime\prime(0)}_{p}(\omega)=\frac{4t_{p}^{2}\omega\omega_{p}\gamma_{p}}{(\omega^{2}-\omega_{p}^{2})^{2}+2\gamma_{p}^{2}(\omega^{2}+\omega_{p}^{2})+\gamma_{p}^{4}}\,,$
(1)

where $t_{p}$ is the light-scattering vertex for the phonon, $\omega_{p}$ is
the mode’s bare frequency, and $\gamma_{p}$ is the half width at half maximum
(HWHM) which is related to the oscillator lifetime.

An exciton in a gapped system (a semiconductor or insulator) is expected to
have a sharply peaked response at the exciton energy described by a bosonic
response function similar to the Eq. (1). The most important difference for a
semimetal is the presence of a gapless continuum of interband particle-hole
excitations enabling the decay of the ${\bf q}=0$ exciton into unbound
particle-hole pairs (Landau damping). To describe the continuum in data, we
find it sufficient to assume a purely relaxational dynamics (corresponding to
strong overdamping) for the excitonic response. This leads to the form

$\chi^{\prime\prime(0)}_{e}(\omega)=\frac{t_{e}^{2}\omega}{(\omega_{e}^{2}/\gamma_{e})^{2}+\omega^{2}}~{},$
(2)

where $t_{e}$ controls the overall intensity determined by the light-
scattering vertex and the excitonic density of states, $\omega_{e}$ is the
frequency of the over-damped excitation, and $\gamma_{e}$ is the relaxation
rate, typically much larger than $\omega_{e}$. Note that for the purpose of
the fitting description below $\Omega_{e}(T)=\omega_{e}^{2}/\gamma_{e}$ is a
single parameter denoting the temperature-dependent peak frequency in the
broad response function $\chi^{\prime\prime}_{e}(\omega)$. For a purely
excitonic transition this peak represents the soft mode:
$\Omega_{e}(T_{c}^{el})=0$ is the condition for bare excitonic transition
temperature at $T_{c}^{el}$, where the static excitonic susceptibility
$\chi_{e}(0)$ diverges (see Eq. (54)).

In the absence of the exciton-phonon interaction between the modes the
respective phononic and excitonic responses [Eqs.(1) and (2)] simply sum up.
The exciton-phonon interaction, however, couples these dynamical responses: in
particular, a bilinear coupling of the exciton and the B2g phonon coordinates
is expected [18] (note that the bare phonons are diagonal normal modes and
thus no bilinear interaction between them is present). The resulting response
can be obtained by solving the coupled equations of motion [Appendix B]. Each
of the responses $\chi^{\prime\prime}_{p}(\omega)$ and
$\chi^{\prime\prime}_{e}(\omega)$ get renormalized by the interaction, and in
addition, the response of the phonon coordinate to the exciton one,
$\chi^{\prime\prime}_{int}(\omega)$, becomes finite. The latter leads to an
essential interference term in the total Raman response function [45, 46, 47].

To appreciate the effect of renormalization, we first consider a simplified
case in which the excitonic continuum couples to a single sharp phonon mode
($\gamma_{p}\ll\omega_{p}$) [Appendix C]. Then, solving the coupled dynamical
equations, one obtains the full Raman response function

$\chi^{\prime\prime}(\omega)=\\\
\frac{\omega[2t_{p}v\omega_{p}-t_{e}(\omega^{2}-\omega_{p}^{2})]^{2}}{(\Omega_{e}^{2}+\omega^{2})(\omega^{2}-\omega_{p}^{2})^{2}+4\Omega_{e}v^{2}\omega_{p}(\omega^{2}-\omega_{p}^{2})+4v^{4}\omega_{p}^{2}}$
(3)

could be broken down into a sum of three contributions:

$\chi^{\prime\prime}(\omega)=\chi^{\prime\prime}_{p}(\omega)+\chi^{\prime\prime}_{e}(\omega)+\chi^{\prime\prime}_{int}(\omega),$
(4)

where the first two terms correspond to the phonon response proportional to
square of light coupling vertex $t_{p}^{2}$ and the excitonic continuum
proportional to $t_{e}^{2}$, respectively, while the third one, that is
proportional to the $t_{p}t_{e}$ combination, is the interference term
appearing due to the exciton-phonon coupling with strength $v$ [47].

We first discuss the phonon response in the presence of the coupling with the
excitonic continuum. As we will show now, the coupling to the continuum of
overdamped excitonic excitations renormalizes the bare linewidth to

$\gamma_{p}^{v}=\frac{v^{2}\omega_{p}}{\omega_{p}^{2}+\Omega_{e}^{2}}\,,$ (5)

thus, the bare $\gamma_{p}$ can be neglected altogether 111Note that the
interaction-induced linewidth $\gamma_{p}^{v}$ is often significantly larger
than the bare width $\gamma_{p}$, which is the case for presented data above
$T_{c}$ . Consequently, an analysis of the phonon lineshape that does not
explicitly include the interaction with the continuum (e.g. by simply adding
the intensities of Fano-shaped phonons and the continuum) would lead to an
erroneous large intrinsic phonon linewidth. The renormalized phonon response
is given by

$\chi^{\prime\prime}_{p}(\omega)=\\\
\frac{4t_{p}^{2}\omega\omega_{p}\gamma_{p}^{v}}{\left[\omega^{2}-(\omega_{p}^{2}-2\Omega_{e}\gamma_{p}^{v})\right]^{2}+4\omega_{p}^{2}\gamma_{p}^{v2}+\frac{(\omega^{2}-\omega_{p}^{2})^{3}}{\omega_{p}^{2}+\Omega_{e}^{2}}}.$
(6)

Comparing Eq. (6) to Eq. (1) for the bare phonon, the phonon frequency
renormalization is $\omega_{p}^{2}\to\omega_{p}^{2}-2\Omega_{e}\gamma_{p}^{v}$
222Note that the cubic term in the denominator can be neglected as long as the
frequency is not far from $\omega_{p}$.. On increasing coupling strength $v$,
the phononic response maximum shifts to
$\omega_{p,max}=\omega_{p}-\frac{v^{2}(2\omega_{p}\Omega_{e}-v^{2})}{2(v^{2}\Omega_{e}+\omega_{p}\Omega_{e}^{2}+\omega_{p}^{3})}$.

For the excitonic response $\chi^{\prime\prime}_{e}(\omega)$ the
renormalization effects are most significant. Those are best illustrated by
the low-frequency slope of the excitonic response function
$\left.\frac{\partial\chi^{\prime\prime}_{e}(\omega)}{\partial\omega}\right|_{\omega\to
0}$. In the absence of exciton-phonon interaction the low frequency slope for
the bare response is equal to $t_{e}^{2}/\Omega_{e}^{2}$ (see Eq. (2)), while
the interaction $v$ is rapidly enhancing the slope as

$\left.\frac{\partial\chi^{\prime\prime}_{e}(\omega)}{\partial\omega}\right|_{\omega\to
0}=\frac{t_{e}^{2}}{{\left[\Omega_{e}(T)-\frac{2v^{2}}{\omega_{p}}\right]^{2}}},$
(7)

leading to a critical value at $v_{cr}=\sqrt{\Omega_{e}\omega_{p}/2}$, when
the excitonic spectral weight is pushed to the lowest frequencies and thus
causing the divergence in the static susceptibility

$\chi(\omega\to
0,T)=\frac{t_{e}^{2}-4t_{e}t_{p}\frac{v}{\omega_{p}}+2t_{p}^{2}\frac{\Omega_{e}(T)}{\omega_{p}}}{\Omega_{e}(T)-\frac{2v^{2}}{\omega_{p}}},$
(8)

signifying the enhancement of the excitonic phase transition temperature,
caused by coupling to the optical phonon.

Finally, the mutual response of the exciton and phonon coordinates appearing
due to the coupling $v$ between them, leads to the sign-changing interference
term

$\chi^{\prime\prime}_{int}(\omega)=\frac{4t_{e}t_{p}v\omega\omega_{p}\frac{\omega_{p}^{2}-\omega^{2}}{\Omega_{e}^{2}+\omega_{p}^{2}}}{\left[\omega^{2}-(\omega_{p}^{2}-2\Omega_{e}\gamma_{p}^{v})\right]^{2}+4\omega_{p}^{2}\gamma_{p}^{v2}+\frac{(\omega^{2}-\omega_{p}^{2})^{3}}{\omega_{p}^{2}+\Omega_{e}^{2}}}.$
(9)

The sign of this term depends on phase difference between the exciton and
phonon oscillators. Because the phase of driven by light phononic oscillator
is flipping to the opposite one at the resonant frequency, the sign of this
term changes close to the bare phonon frequency $\omega_{p}$. Thus, this term
is chiefly responsible for a skewed, asymmetric shape of the resulting Fano
feature. Depending on the sign of $v$ the peak is skewed to the left or to the
right of the original phonon frequency. Note that even for weak coupling
$v\ll\omega_{p}$ this term can have an appreciable magnitude close to the
phonon energy.

To analyze the actual measured low-frequency data shown in Fig. 6, we use the
model with three B2g optical phonon modes, all interacting with the excitonic
continuum, see for details Appendix B. As above, the total Raman response
$\chi^{\prime\prime}(\omega)$ can be similarly decomposed into three
contributions, Eq. (4): the first two describing the renormalized phononic and
excitonic responses, respectively, with the third one arising due to the
interference effects. In Fig. 7, an example of such decomposition is shown for
the data taken at 380 K, along with the deduced bare responses (without the
effect of coupling), $\chi^{\prime\prime(0)}_{p}(\omega)$ and
$\chi^{\prime\prime(0)}_{e}(\omega)$. The renormalized phonon features become
broader due to the interaction with the continuum and their frequencies shift.
Most importantly, the coupling noticeably increases the excitonic response at
low frequencies. The sign-changing interference term enhances the asymmetric
lineshape of the combined response.

Figure 7:  The effect of coupling on Raman spectrum, relevant to the data
measured for Ta2NiSe5 at 380 K. The Raman response functions are shown for
bare (dashed lines) and renormalized by interaction (solid lines),
$\chi^{\prime\prime(0)}_{e,p}(\omega)$ and
$\chi^{{}\prime\prime}_{e,p}(\omega)$ correspondingly. The interference term
$\chi^{\prime\prime}_{int}(\omega)$ is shown by solid orange line. The solid
red line corresponds to the total sum of the solid blue, solid green, and
solid orange lines, Eq. (4). The black dots represent the experimentally
measured spectrum.

To fit the $ac$ spectra, first, we determine the $t_{e}$ parameter of the
excitonic continuum (Eq. (2)) by fitting the phonon free region of spectra
between 25 and 40 meV. As expected, this quantity remains temperature-
independent [Fig. 10(b)]. Second, we perform a global fitting of the spectral
region below 22 meV for all measured temperatures. We assume (consistently
with the expectation due to anharmonic effects, see Eq. (16-17) below) the
linewidth of the three B2g-symmetry phonon modes to be a linear function of
temperature above $T_{c}$ . For spectra below $T_{c}$ , we fit the ”leaked”
phonon modes with the Lorentzian lineshapes, and account for their
contribution in the global fitting.

###  V.3 The coupling between excitonic continuum and acoustic modes

Besides the optical phonon modes, the excitonic continuum also couples to the
acoustic phonon modes. The acoustic branch has a linear dispersion
$\omega_{s}(q)=c_{s}q$ ($c_{s}$ is sound velocity) at small wavectors $q$; its
coupling to light and to the excitonic continuum both vanish in the long-
wavelength limit [50]: $t_{s}=\tau_{s}\sqrt{q}$ and $v_{s}=\beta_{s}\sqrt{q}$
($\tau_{s}$ and $\beta_{s}$ are constants). However, the acoustic mode has
important contribution to the enhancement of the phase-transition temperature
[51]. The the diverging condition for static Raman susceptibility
$\chi(\omega\to 0,T)$ is defined by coupling to all modes, optical and
acoustic:

$\tilde{\Omega}_{e}(T_{c})-\frac{2\beta_{s}^{2}}{c_{s}}=0,$ (10)

in which

$\tilde{\Omega}_{e}(T)=\Omega_{e}(T)-\sum_{i=1}^{3}\frac{2v_{i}^{2}}{\omega_{pi}}.$
(11)

This equation determines the symmetry-breaking phase transition temperature
$T_{c}$ .

Because the spectroscopic Fano feature associated with the coupling between
the long-wavelength longitudinal acoustic mode and the excitonic continuum
lies at frequencies far below accessibility of Raman experiment, the constant
$\beta_{s}$ cannot be derived directly from an interference feature of a
typical Raman spectra. Nevertheless, the coupling constant $\beta_{s}$ has
observable consequences already above $T_{c}$. In particular, the acoustic
mode has been shown to soft on cooling towards $T_{c}$ [52]. This softening
does not result from intrinsic instability of the acoustic modes; rather, it
is caused by the coupling of the acoustic mode to the softening excitonic
excitations [Eq. (2)].

To demonstrate this point, we solve for the lowest-energy pole of the Green’s
function for the interacting phononic and excitonic excitations assuming an
infinitesimal $q$. The energy of the pole gives the renormalized
(experimentally-measured) sound velocity $\tilde{c}_{s}$ above $T_{c}$ [Eq.
(57) of Appendix B]:

$\tilde{c}_{s}^{2}=c_{s}^{2}-\frac{2\beta_{s}^{2}c_{s}}{\tilde{\Omega}_{e}(T)},$
(12)

in which $c_{s}$ is the bare sound velocity in the absence of coupling to the
excitons. Eq. (10) enables us to rewrite Eq. (12) as

$\tilde{c}_{s}(T)=c_{s}\sqrt{1-\frac{\tilde{\Omega}_{e}(T_{c})}{\tilde{\Omega}_{e}(T)}}.$
(13)

From Eq. (13), it is clear that the sound velocity softens to zero at $T_{c}$
in the absence of any intrinsic ferroelastic instability. The renormalization
of the sound velocity above $T_{c}$ can be estimated by using Eq. (13). Using
the low-temperature value of the sound velocity $c_{s}$ [52], we can then
calculate $\tilde{c}_{s}(T)$ with the parameters obtained from the Fano fits,
i.e. without free parameters. The result yields
$\tilde{c}_{s}(T=400K)/c_{s}\approx 0.65$, which is in remarkable agreement
with the experimental ratio $\tilde{c}_{s,exp}(T=400K)/c_{s,exp}\approx 0.65$.
The agreement between the estimated and experimentally-determined values
suggests that in Ta2NiSe5, the softening of the acoustic mode results solely
from the coupling to the softening excitonic excitations.

###  V.4 The generalized Fano model for data below $T_{c}$

Below $T_{c}$ , especially around 300 K, this model appears to be insufficient
to account for an additional low-frequency spectral weight, see Fig. 6(e) and
Fig. 8. To understand the origin of this spectral feature below a few meV, we
recall that below $T_{c}$ formation of quasi-periodic structural domains have
been observed by transmission electron microscopy (TEM) [9, 15]. Such quasi-
periodic structure can take a recoil of quasi-momenta, enabling Raman coupling
to acoustic mode at finite momentum $q_{d}=\frac{2\pi}{d}$, where $d$ is the
periodicity of the domain pattern, causing the appearance of the additional
spectral feature at frequency $\omega_{d}\approx c_{s}q_{d}$. Therefore,
fitting this ultra-low-frequency spectral feature provides an independent
approach to determine the coupling constant $v_{d}=\beta_{s}\sqrt{q_{d}}$
between the acoustic lattice excitations and the excitonic continuum, and
hence define the constant $\beta_{s}$ that controls the enhancement of the
transition temperature to $T_{c}$ due to coupling between excitons and
longitudinal strain fields of the same symmetry. This low-energy feature due
to recoil on the quasi-periodic structure of domain walls is best seen about
30 K below $T_{c}$ , when the excitonic continuum is sufficiently suppressed
but still adequate to provide a Fano-interference feature.

Figure 8:  The contribution from the acoustic modes to low-frequency Raman
spectrum in $ac$ polarization, relevant to the data measured for Ta2NiSe5 at
300 K. The data is shown by black dots; the total Fano fit is shown as red
curve; the bare excitonic and phononic components are presented in blue and
green, respectively; the coupled response of the acoustic and excitonic
components is shown in purple.

In Fig. 9 we compare two TEM images from $ac$ plane of Ta2NiSe5 measured above
and below $T_{c}$ . The domains have stripe shape, aligned parallel to the
$a$-axis. The average spacing $\bar{d}$ between these stripes along $c$-axis
direction is on the order of 200 $\AA$. For the purpose of spectral analysis,
we will assume that the spacing between stripes $i$ in units of the crystal
unit cell constant $c$ follows the Poisson distribution

$P(i;\mu=\frac{\bar{d}}{c})=\frac{\mu^{i}e^{-\mu}}{i!},$ (14)

where $\mu$ is average inter-domain distance in the number of $c$-direction
unit cells.

Figure 9:  Two images of the Ta2NiSe5 recorded from the $ac$ plane by
transmission electron microscopy at (a) 350 K and (b) 300 K.

We deduce the unrenormalized speed of sound from the dispersion of the
acoustic mode measured by inelastic x-ray scattering at temperatures far away
from $T_{c}$ : $c_{s}\approx 30$ meV Å, see Ref. [52]. We also note that at
high temperatures the acoustic excitations away from long-wavelength limit are
significantly broadened [52].

Recognizing that the low-frequency feature is inhomogeneously broadened by
random distribution of the stripe distances, for the fitting procedure to the
measured response function we perform summation over the distribution
function:

$\chi^{\prime\prime}(\omega)=\sum_{i=1}^{\infty}P(i;\mu)\chi^{\prime\prime}_{i}(\omega),$
(15)

in which each $\chi^{\prime\prime}_{i}(\omega)$ is the full response function
containing coupling to acoustic mode for domain walls $i$ lattice constants
apart, see Eq. (51) in Appendix. As a reminder, for each individual component,
the frequency and HWHM are proportional to $q_{i}$; the light-scattering
vertex and coupling are proportional to $\sqrt{q_{i}}$.

###  V.5 The fitting results for Ta2NiSe5

The fits to the data, as well as the phononic and excitonic components, are
shown in Fig. 6. For the 300 K and 265 K spectra below $T_{c}$ we also include
the acoustic contribution due to coupling via quasi-periodic structure of
domain walls. In Fig. 8 we show the acoustic contribution at 300 K. The
coupling of the excitonic continuum to the acoustic components enhances the
low-energy response. The parameter $\beta=7.7$ meV Å${}^{\frac{1}{2}}$ is
consistent with Eq. (10) for $T_{c}$ .

The spectral parameters: frequency, FWHM and integrated intensity, – of the
three B2g phonon modes are shown in SubSection. VI.2. The rest of the fitting
parameters are given in Fig. 10.

The temperature dependence of the coupling strength between the individual
phonon modes and the excitonic continuum are detailed in Fig. 10(a). The
ratios of the coupling strength to the phonon frequency for the
B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$ modes above $T_{c}$ are 0.32 and -0.34,
respectively. These ratios are an order of magnitude larger than the typical
values for stable systems with similar phonon frequency [53, 54, 47]. On the
contrary, the exciton-phonon interaction with the B${}_{2g}^{(3)}$ mode is
weak. The signs of the coupling for the B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$
modes are opposite. For these two modes, the magnitude of coupling is
temperature independent above $T_{c}$ , but decreases and saturates on cooling
below $T_{c}$ . The temperature dependence of the coupling strength is
influenced mainly by two factors: the screening effect of free carriers,
reduced below $T_{c}$ due to the emergence of a pronounced spectral gap [55,
23], and the change of the crystal structure below $T_{c}$.

Figure 10:  Temperature dependence of the Fano fitting parameters for the $ac$
Raman spectra of Ta2NiSe5. (a) The coupling strength $v_{i}$ (i=1,2,3) in Eq.
(38). (b) The light-scattering vertex $t_{e}$, and the frequency at which the
excitonic continuum has maximum intensity
$\Omega_{e}(T)=\omega_{e}^{2}/\gamma_{e}$, Eq. (2). The blue line is a linear
fit to the $\Omega_{e}(T)$ data above $T_{c}$ . The dashed lines indicate
transition temperature $T_{c}$ and bare excitonic transition temperature
$T_{c}^{ex}$ .  Figure 11: The calculated vibrational patterns corresponding
to the three B2g-symmetry phonon modes of Ta2NiSe5.

The difference in magnitude of the exciton-phonon coupling for
B${}_{2g}^{(1,2)}$ and B${}_{2g}^{(3)}$ modes can be readily understood from
the corresponding displacement patterns obtained from the DFT calculations for
Ta2NiSe5 and shown in Fig. 11. The calculated patterns are consistent with
those reported in other studies [52, 20]. While both the B${}_{2g}^{(1,2)}$
modes involve displacements of Ta and Se, the B${}_{2g}^{(3)}$ mode involve a
displacement of Se almost exclusively (the Ni atoms are at the inversion
centers and thus do not contribute to the Raman-active phonon modes). As the
electronic bands in Ta2NiSe5 close to the Fermi level are believed to be
predominantly formed by Ta and Ni electrons [18, 19, 12], with much smaller
contribution of Se, it appears natural that the B${}_{2g}^{(3)}$ mode does not
couple to the low-energy electronic degrees of freedom. Furthermore, the
vibration of the Ta atoms for B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$
displacements are in anti-phase, which can explain the opposite signs of the
exciton-phonon coupling for these modes.

The temperature dependence of the light-scattering vertex $t_{e}$ and the
frequency $\Omega_{e}$ at which the continuum has maximum intensity are shown
in Fig. 10(b). Above $T_{c}$ , the quantity $\Omega_{e}$ linearly decreases on
cooling; below $T_{c}$ , it rapidly increases and saturates at low
temperature.

###  V.6 The fitting results for Ta2Ni(Se1-xSx)5 with $x$ = 0.25 and 0.67

To compare with the results for Ta2NiSe5, we also fit the $ac$ polarization
spectra for the alloy compositions ($x$ = 0.25 and 0.67) above their
transition temperature. We are not able to perform a Fano analysis below
$T_{c}$ for two reasons: (a) we have no knowledge of the domain structure and
the sound velocity; and (b) we cannot properly subtract the leakage of the Ag
modes. In Fig. 12 we show one characteristic Fano fit for Ta2Ni(Se0.75S0.25)5
and Ta2Ni(Se0.33S0.67)5, respectively. The fitting parameters are given in
Fig. 13.

Figure 12:  Raman response $\chi^{\prime\prime}$, represented by black dots,
in the ac scattering geometry for Ta2Ni(Se0.75S0.25)5 and Ta2Ni(Se0.33S0.67)5
with the Fano fits [Eq. (36)] shown as red curves. The spectral resolution is
0.19 meV. The phonon modes [Eq. (1)] and excitonic continuum [Eq. (2)] are
represented by green and blue curves, respectively. Figure 13:  Temperature
dependence of the Fano fitting parameters for the ac Raman spectra of
Ta2Ni(Se0.75S0.25)5 (a-b) and Ta2Ni(Se0.33S0.67)5 (c-d). (a,c) The coupling
strength $v_{i}$ (i=1,2,3) in Eq. (38). (b,d) the light-scattering vertex
$t_{e}$, and the energy at which the excitonic continuum has maximum intensity
$\Omega_{e}=\omega_{e}^{2}/\gamma_{e}$ of the overdamped electronic mode, Eq.
(40). The blue line is a linear fit to the $\Omega_{e}$ data.

The results are similar to the case of Ta2NiSe5: for the coupling strength, we
find positive $v_{1}$, negative $v_{2}$, and negligible $v_{3}$; the light-
scattering vertex $t_{e}$ has no temperature dependence above the transition
temperature; the energy at which the excitonic continuum has maximum
intensity, $\Omega_{e}=\omega_{e}^{2}/\gamma_{e}$, decreases linearly on
cooling above the transition temperature. Extrapolating to zero value of
$\Omega_{e}$, we obtain $55\pm 18$ K for Ta2Ni(Se0.75S0.25)5 and $-400\pm 70$
K for Ta2Ni(Se0.33S0.67)5.

In Table. 4 we present the doping dependence of the relevant physical
quantities above the transition temperature for Ta2Ni(Se1-xSx)5 family.
Because the coupling strength increases with sulfur doping, determining the
intrinsic phonon parameters, especially the linewidth, becomes more difficult
for Ta2Ni(Se0.75S0.25)5 and Ta2Ni(Se0.33S0.67)5.

Table 4: The doping dependence of the scattering vertex and the coupling above the transition temperature for the interaction between the excitonic continuum and phonon modes in the B2g scattering geometry of Ta2Ni(Se1-xSx)5 family. The quantities $t_{e}$ is the vertex of light scattering process for the excitonic continuum. The quantities $v_{i}$ (i=1,2) are the coupling between the excitonic continuum and the B${}_{2g}^{(i)}$ phonon mode. | x=0 | x=0.25 | x=0.67
---|---|---|---
$t_{e}$ (arb. units) | 7.22 | 7.36 | 7.48
$v_{1}$ (meV) | 2.45 | 2.61 | 3.61
$v_{2}$ (meV) | -4.03 | -4.61 | -5.87

##  VI Temperature dependence of the intrinsic phonon parameters

In this section we present the temperature dependence of intrinsic phonon-mode
parameters: energies, intensities and linewidths. The results for B2g-symmetry
modes are given in SubSec. VI.1, and those for Ag-symmetry modes are given in
SubSec. VI.2.

###  VI.1 B2g-symmetry Phonon Modes

####  VI.1.1 Ta2NiSe5

Figure 14:  Temperature dependence of the spectral parameters: the bare
frequency, FWHM, and the integrated intensity, – for the B2g-symmetry phonon
modes of Ta2NiSe5. For each mode, the left panel presents the frequency and
FWHM; and the right panel presents the integrated intensity in the allowed and
’forbidden’ scattering geometries. The solid lines represent the fits to the
anharmonic decay model [Eqs. (16-17)].

In Fig. 14 we show the temperature dependence of the spectral parameters for
B2g-symmetry phonon modes in Ta2NiSe5: the bare phonon frequency, FWHM, and
integrated intensity.

Below $T_{c}$ , the temperature dependence of both frequency $\omega_{p}(T)$
and FWHM $\Gamma_{p}(T)=2\gamma_{p}(T)$ of the phonon modes can be accounted
for by standard model assuming anharmonic decay into two phonons with
identical frequencies and opposite momenta [56]:

$\omega_{p}(T)=\omega_{0}-\omega_{2}[1+\frac{2}{e^{\hbar\omega_{0}/2k_{B}T}-1}],$
(16)

and

$\Gamma_{p}(T)=\Gamma_{0}+\Gamma_{2}[1+\frac{2}{e^{\hbar\omega_{0}/2k_{B}T}-1}].$
(17)

We note that a low value of $\Gamma_{0}$ indicated high quality of the
crystals.

Above $T_{c}$ , however, the slope of the FWHM increases for the three phonon
modes, and the FWHM becomes larger than the value predicted by the anharmonic
decay model. The appearance of additional decay channels above $T_{c}$ is
related to the presence of gapless particle-hole pairs. Indeed, above $T_{c}$
Ta2NiSe5 is gapless spectroscopically[23], while below $T_{c}$ a gap rapidly
develops at low energies, suppressing the damping due to particle-hole pairs.
Because the Fano model used for fitting takes into account the damping due to
exciton-phonon interaction, existence of additional damping above $T_{c}$
suggests the importance of the interactions beyond that model, i.e. nonlinear
ones.

More noticeable is that the frequencies of the B${}_{2g}^{(1)}$ and
B${}_{2g}^{(2)}$ modes exhibit a large increase on cooling from around 350 K
to $T_{c}$ . The frequency of the B${}_{2g}^{(3)}$ mode, however, does not
show such anomaly. Only B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$ modes exhibit
this anomaly; interestingly (1) they are allowed to couple to interband
scattering by symmetry selection rules and (2) their corresponding vibration
involve motion of Ni atoms and hence they couple to the electronic bands near
Fermi level. Although B${}_{2g}^{(3)}$ mode can also couple to the interband
scattering, its vibrational pattern almost does not contain motion of Ni
atoms.

As for the integrated intensity, B${}_{2g}^{(1)}$ and B${}_{2g}^{(3)}$ modes
have around 2-fold increase of intensity on cooling below $T_{c}$ , while
B${}_{2g}^{(2)}$ mode shows temperature-independent intensity. Two factors
could influence the intensity. First, the interaction-induced gap opening up
below $T_{c}$ should play a major role. The system gradually changes from a
semimetal at high temperature to an insulator at low temperature. The
screening effect, which reduces the light-scattering vertex, and in turn, the
intensity for phonon modes, is suppressed on cooling. This factor therefore
favors increase of intensity on cooling. Second, the structural change below
$T_{c}$ could play also a role. The ion positions are shifted within the unit
cell below $T_{c}$ , and the vibrational patterns of the three B2g modes are
modified. Therefore, the polarizability induced by lattice vibrations, which
is proportional to the phonon intensity, also change with temperature. The
fact that B${}_{2g}^{(2)}$ mode shows temperature-independent intensity might
be related to its unique vibrational pattern.

####  VI.1.2 The alloy compositions ($x$=0.25, 0.67)

In Fig. 15 and 16 we show the temperature dependence of the B2g-mode spectral
parameters for Ta2Ni(Se0.75S0.25)5 and Ta2Ni(Se0.33S0.67)5, respectively.
Although we do not perform Fano analysis for the data below $T_{c}$ , we use
Lorentzian lineshape to fit the B${}_{2g}^{(1)}$ mode at low-enough
temperatures, at which the lineshape is essentially symmetric. For the
B${}_{2g}^{(2)}$ and B${}_{2g}^{(3)}$ modes, because they are not well
separated from the leakage of Ag modes, it is difficult to reliably obtain
their spectral parameters.

Figure 15:  Temperature dependence of the spectral parameters: frequency,
FWHM, and integrated intensity, – for the B2g-symmetry phonon modes of
Ta2Ni(Se0.75S0.25)5. For each mode, the left panel presents the energy and
FWHM; the right panel presents the integrated intensity in the allowed and
’forbidden’ scattering geometries. Figure 16:  Temperature dependence of the
spectral parameters: frequency, FWHM, and integrated intensity, – for the
B2g-symmetry phonon modes of Ta2Ni(Se0.33S0.67)5. For each mode, the left
panel presents the energy and FWHM; the right panel presents the integrated
intensity in the allowed and ’forbidden’ scattering geometries.

For the alloy compositions, the increase of the FWHM slope above $T_{c}$ is
not as strong as in the case of Ta2NiSe5. Moreover, although for Ta2NiSe5 the
large frequency increase of the B${}_{2g}^{(1)}$ mode on cooling happens in a
20 K energy range, the frequency increase for the alloy compositions happens
in a much larger energy range.

####  VI.1.3 Ta2NiS5

Ta2NiS5 is a semiconductor that, in contrast to Ta2NiSe5 , does not show
signatures of an excitonic insulator like the flattening of valence band
dispersion [57, 21]. Consistent with its semiconductor nature, we observe no
excitonic continuum and in turn no asymmetric lineshape for the phonon modes,
Fig. 2(c-d). In Fig. 17 we zoom in on the temperature dependence of the
phonons in $ac$ scattering geometry. The three B2g-symmetry phonon modes
exhibit conventional Lorentzian lineshapes, and the linewidth has almost
4-fold decrease on cooling from 315 K to 35 K. Moreover, the B${}_{2g}^{(1)}$
and B${}_{2g}^{(2)}$ modes show softening behavior on cooling.

Figure 17:  Temperature dependence of B2g phonons in the $ac$ scattering
geometry for Ta2NiS5. The spectral resolution is 0.19 meV for the 315 K data,
and 0.06 meV for the other data.

In Fig. 18 we show the temperature dependence of the spectral parameters for
the B2g-symmetry phonon modes for Ta2NiS5: the frequency, FWHM, and the
integrated intensity. These spectral parameters are obtained by fitting the
measured spectral features with Lorentzian lineshapes.

Figure 18:  Temperature dependence of the spectral parameters: frequency,
FWHM, and integrated intensity, – for the B2g-symmetry phonon modes of
Ta2NiS5. For each mode, the left panel presents the energy and FWHM; the right
panel presents the integrated intensity in the allowed and ’forbidden’
scattering geometries.

The behavior of the B2g-symmetry modes is not consistent with the anharmonic
decay model: for B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$ modes, the frequency
anomalously decreases on cooling; for B${}_{2g}^{(3)}$ mode, the decrease of
FWHM on cooling is too steep to be accounted by the anharmonic decay model.
Hence, we suggest that there is a change in the phonon self energy, which must
affect the the apparent mode frequency. For the B${}_{2g}^{(1)}$ and
B${}_{2g}^{(2)}$ modes, whose frequency is below 30 meV, the energy decreases
on cooling; for the B${}_{2g}^{(3)}$ mode, whose frequency is above 30 meV,
the energy increases on cooling.

The integrated intensity of B${}_{2g}^{(2)}$ mode is temperature-independent,
while that of the other modes increases on cooling. However, different from
the case of Ta2NiSe5, in which the increase of the intensity happens below
$T_{c}$ , the intensity increase of the Ta2NiS5 modes is through the whole
measured temperature range.

###  VI.2 Ag-symmetry phonon modes

In this subsection we discuss the properties of the full-symmetric phonon
modes; the relevant results are shown in Fig. 19 with respect to the overall
spectral weight distribution in the Ag channel, and in Figs. 20 and 21
regarding the spectral parameters of the individual Ag modes. In Fig. 21 we
compare the temperature dependence of the spectral parameters (frequency,
FWHM, and integrated intensity) of the Ag-symmetry phonon modes for Ta2NiSe5
and Ta2NiS5 crystals. The temperature dependence of both frequency and FWHM
above $T_{c}$ for these modes can be accounted by the anharmonic decay model
[Eq. (16-17)].

The FWHM of the Ag-symmetry modes (except for A${}_{g}^{(3)}$ and
A${}_{g}^{(8)}$) for Ta2NiSe5 exhibit an anomalous increase above the
transition temperature, which we attribute to enhanced electron-phonon
scattering rate in its semimetal phase. However, the modes of Ta2NiS5 show no
anomaly of FWHM, because it is a semiconductor with a direct gap observed
throughout the measured temperature range.

The intensity of most modes has more than 2-fold increase on cooling, with a
few exceptions: for Ta2NiSe5, the intensity of the A${}_{g}^{(1)}$ mode is
independent of temperature, while the intensity of the A${}_{g}^{(2)}$ phonon
mode decreases on cooling; for Ta2NiS5, the intensities of the A${}_{g}^{(1)}$
and A${}_{g}^{(2)}$ modes are temperature independent.

The enhancement of intensity on cooling in the semimetallic samples could in
principle be related to the change of electronic structure (less screening
effect) below $T_{c}$. However, an interesting insight can be further obtained
by computing the integral
$I_{aa}(\omega)=\int_{0}^{\omega}\frac{\chi_{aa}^{\prime\prime}(\omega^{\prime})}{\omega^{\prime}}\,d\omega^{\prime}$
(Fig. 19). In the limit $\omega\to\infty$ this integral is proportional to the
static susceptibility in the $aa$ scattering geometry, which is expected to be
approximately constant as a function of temperature since there is no
instability in Ag channel. On cooling, the phonon intensity is enhanced [Fig.
21], while the electronic continuum is suppressed [Fig. 2(a)]. However, the
integral $I_{aa}$ is essentially temperature independent at 70 meV. This
conservation implies that the enhancement of the phonon intensity is balanced
by the reduction of the electronic intensity. Note that these effects occur at
temperatures below $T_{c}$, where the continuum in B2g symmetry is strongly
suppressed, indicating the opening of a gap. On the other hand, strong
correlation effects have been shown to lead to a non-zero intensity within the
gap [23]. Thus, the peculiar balance between the electronic and phononic
contributions at low energies implies an unconventional coupling of the
electronic modes to the Ag phonons in the correlated excitonic insulator
state.

Figure 19: The integral
$I_{aa}(\omega)=\int_{0}^{\omega}\frac{\chi_{aa}^{\prime\prime}(\omega^{\prime})}{\omega^{\prime}}\,d\omega^{\prime}$
as a function of the Raman shift $\omega$ for Ta2NiSe5 in $aa$ scattering
geometry. Figure 20:  Temperature dependence of the spectral parameters:
frequency, FWHM, and integrated intensity, – for the A${}_{g}^{(1)}$ phonon
mode of Ta2Ni(Se1-xSx)5 with $x=0.25$ (a-b) and 0.67 (c-d). For each sample,
the left panel presents the energy and FWHM; the right panel presents the
integrated intensity. The solid lines represent the fits to the anharmonic
decay model [Eqs. (16-17)]. Figure 21:  Temperature dependence of the spectral
parameters: frequency, FWHM, and integrated intensity, – for the Ag-symmetry
phonon modes for Ta2NiSe5 and Ta2NiS5. For each mode, the left panel presents
the energy and FWHM; the right panel presents the integrated intensity. The
solid lines represent the fits to the anharmonic decay model [Eqs. (16-17)].

For comparison, in Fig. 20 we show the temperature dependence of the spectral
parameters for the A${}_{g}^{(1)}$ mode for Ta2Ni(Se1-xSx)5 with $x=0.25$ and
0.67. Because of the difficulty caused by broad lineshape and two-frequency
behavior of the phonon modes for alloy compositions, only the lowest-frequency
A${}_{g}^{(1)}$ mode, which are well separated from other modes in frequency,
renders reliable fitting results. For Ta2Ni(Se0.75S0.25)5, some linewidth
broadening is observed above $T_{c}$ , which is weaker than for Ta2NiSe5.
However, for Ta2Ni(Se0.33S0.67)5 almost no broadening is observed.

##  VII Relation to time-resolved experiments

In this section we convert Raman response from frequency domain to time
domain, and discuss relevant time-resolved studies on Ta2NiSe5.

Figure 22:  Time-domain signal for the B2g-symmetry excitations of Ta2NiSe5.
(a) The Raman response for Ta2NiSe5 measured in ac geometry at 380 K. The Fano
fit [Eq. (35)] is shown as the red curve; the bare phonon modes [Eq. (1)] are
shown by the green curve; the bare excitonic continuum [Eq. (2)] is shown by
the blue curve. The time-domain signal corresponding to the Fano fit, bare
phonon modes, and bare excitonic continuum, obtained by inverse Fourier
transform, are shown in (b), (c), and (d), respectively. (e) The Raman
response for Ta2NiSe5 measured in ac geometry at 35 K. The corresponding time-
domain signals are shown in (f), (g), and (h), respectively.

We first discuss the appearance of B2g-symmetry excitations in the time
domain, obtained by inverse Fourier transform of the Raman response, see Fig.
22. Above the transition temperature, the phonon-exciton interaction
significantly broadens the spectral features [Fig. 22(a)]; as a result the
deduced time-resolved response decays fast and essentially dies out before 5
ps [Fig. 22(b)]. For the bare phononic response, the time-domain signal beyond
4 ps is dominated by the oscillation of 2 THz mode [Fig. 22(c)], corresponding
to the B${}_{2g}^{(1)}$ phonon. The signal of the bare excitonic response
shows a pure relaxational behavior [Fig. 22(d)]. Comparing Fig. 22(b) and (c),
we note that it is difficult to identify the Fano interference feature from
time-domain signal, because such interference exhibits no distinct characters
in the time domain, except for that the oscillations are strongly damped.
Indeed, in ultrafast studies, the interference nature is not revealed even
after Fourier transform of the time-domain data to the frequency domain [17,
16, 20]. The likely reason for that is the high pump fluence used, causing the
exciton and phonon responses decouple already on the very early time scales.

Below the transition temperature, the energy at which the continuum has
maximum response moves to higher energy, and the phonon-exciton coupling
strength is reduced. Consequently, the interference effect is suppressed and
the combined response is essentially the same as the bare phononic response
[Fig. 22(f-g)]. The bare excitonic response still has a pure relaxational time
dependence, with very short lifetime [Fig. 22(d) and (h)].

For comparison, in Fig. 23 we present the time-domain signal for the Ag-
symmetry excitations. Above the transition temperature, although multiple
oscillations are simultaneously present at short time scale, beyond 10 ps the
time-domain signal is dominated by the oscillation of 1 THz [Fig. 23(b)],
corresponding to the A${}_{g}^{(1)}$ phonon mode whose spectral width is much
smaller than others [Fig. 23(a)]. At low temperature, the difference between
the spectral widths of various modes is reduced [Fig. 23(c)]; hence, these
modes have comparable lifetime and multiple oscillations survive beyond 10 ps
[Fig. 23(d)].

Figure 23:  Time-domain signal for the Ag-symmetry excitations of Ta2NiSe5.
(a) The Raman response for Ta2NiSe5 measured in aa geometry at 380 K. The
time-domain signal, obtained by inverse Fourier transform of the data, is
shown in (b). (c) The Raman response for Ta2NiSe5 measured in aa geometry at
35 K. The corresponding time-domain signal is shown in (d).

In relation to the ultra-fast studies of Ta2NiSe5 [17, 58, 59, 15], the Raman
results are consistent with the ultra-fast data under low fluence. One ultra-
fast work [17] provides evidence for coupling between the A${}_{g}^{(1)}$ mode
and the amplitude mode of the condensate under high-fluence pumping. We note
that in our studies we can unambiguously identify the amplitude mode with the
excitonic continuum we observe. Above the transition temperature, the
overdamped excitonic mode softens on cooling towards $T_{c}$ ; just below the
transition temperature, its energy increases on further cooling. Such
temperature dependence of mode energy is characteristic of an amplitude mode
while the optical B2g phonons only harden on cooling. In Fig. 24 we present
the time-domain signal corresponding to the excitonic mode around the
transition temperature.

Figure 24:  Time-domain signal for the excitonic continuum of Ta2NiSe5 above
and below the transition temperature, plotted in semi-log scale. The data
above $T_{c}$ are dashed for clarity. The relaxation time starts to decrease
just below $T_{c}$ . Inset: temperature dependence of the energy at which the
excitonic continuum has maximum intensity, $\Omega_{e}(T)$.

Additionally, a mode associated with the phase of the excitonic order
parameter may exist. The electron-phonon coupling and exchange interactions
modify the otherwise sombrero-shape free-energy landscape, reducing the
continuous U(1) symmetry to a discrete Z2 symmetry, leading to a finite energy
of the phase mode. A recent pump-probe microscopy study suggests that the
phase-mode energy is smaller than the A${}_{g}^{(1)}$-mode energy (around 4
meV) [58]. However, we do not observe any mode down to 0.4 meV at low
temperature [Fig. 22(e)]. Furthermore, the phase oscillations of the order
parameter should harden above $T_{c}$, which would have manifested itself in
the low-energy range of the spectra. Such signatures are not observed, either
[Fig. 6]. Therefore, existence of the phase mode below 4 meV is not supported
by the Raman data.

##  VIII Conclusions

We have employed polarization resolved Raman spectroscopy to conduct a
systematic spectroscopic study of the lattice dynamics in Ta2Ni(Se1-xSx)5
($x=0$, …, 1) family of the excitonic insulators.

We identify and classify by symmetry all the Raman active phonon modes of Ag
and B2g symmetries. A change in selection rules is detected at temperature
$T_{c}(x)$, indicating the orthorhombic-to-monoclinic structural phase
transition related to the excitonic insulator state [23, 24]. We find that the
symmetry breaking transition persists for entire Ta2Ni(Se1-xSx)5 family with
$T_{c}(x)$ monotonically decreasing from 328 K for $x=0$ to 120 K for $x=1$.
Its signatures in the resistivity data, however, become weak or undetectable
for large sulfur concentration $x$ that we attribute to the weakness of the
symmetry breaking at large $x$, demonstrated by the decrease of phonon
intensity in the ’forbidden’ scattering geometry.

For $x<0.7$, the two lowest-frequency $ac$-quadrupole-symmetry
B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$ phonon modes show strongly asymmetric
lineshapes at high temperatures, which is indicative of coupling between the
phonons and an excitonic continuum of the same symmetry [23, 24]. Within the
framework of extended Fano model, we develop a quantitative description of the
observed lineshapes, enabling us to disentangle the excitonic and phononic
contributions to the spectra, to derive the intrinsic phonon parameters and
determine the exciton-phonon interaction strength, that affects the transition
temperature $T_{c}(x)$ [24]. The displacement patterns obtained from ab-initio
calculations explain the trends in the deduced exciton-phonon coupling values.
At $T<T_{c}$ the remnant excitonic continuum demonstrates signatures of
coupling to the finite-momentum B2g -symmetry acoustic phonons, allowed due to
scattering on a quasi-periodic structural of domain walls. We have also shown
that the coupling to the excitonic continuum explains the acoustic mode
softening observed in [52, 24].

The intrinsic parameters of the optical phonons for Ta2NiSe5 and Ta2NiS5 at
low temperatures are mostly in good agreement with anharmonic decay model and
DFT calculations. For alloy compositions a two-mode behavior is found for most
modes, where signatures appear at two frequencies, corresponding to the ones
in Ta2NiSe5 and Ta2NiS5. However, several types of anomalous behavior have
been observed in the temperature dependencies. For $x<0.7$, the
B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$ modes anomalously harden close to
$T_{c}$ (e.g., Fig. 14), indicating an interaction with continuum beyond Fano
model description [23]. For Ta2NiS5 (Fig. 18) the B${}_{2g}^{(1)}$ and
B${}_{2g}^{(2)}$ modes do not show Fano shapes, but the frequencies of these
modes anomalously soften on cooling, although never going critical. Finally,
the intensity of the most modes is strongly temperature dependent.
Interestingly, for Ta2NiSe5 we observe that the sum of electronic and phononic
contributions to the $aa$ static susceptibility is conserved (Fig. 19),
pointing to an unexpected electron-phonon coupling mechanism within the
excitonic insulator state.

In addition, based on our results we have provided an interpretation of recent
time-resolved pump-probe experiments and discussed the signatures expected
from the modes we have observed, see Figs. 22, 23 and 24. Overall, this work
provides a comprehensive study lattice dynamics in Ta2Ni(Se1-xSx)5; more
generally, the unconventional behaviors we observed point to the importance of
the effects of electron-phonon coupling in correlated semimetals.

###### Acknowledgements.

M.Y. and P.A.V. contributed equally to this work. We acknowledge discussions
with K. Haule. We are grateful to Y. Kauffmann for help with TEM measurements.
The spectroscopic work conducted at Rutgers was supported by NSF Grant No.
DMR-1709161 (M.Y. and G.B). P.A.V. acknowledges the Postdoctoral Fellowship
support from the Rutgers University Center for Materials Theory. The sample
growth and characterization work conducted at the Technion was supported by
the Israel Science Foundation Grant No. 320/17 (H.L., I.F. and A.K.). H.L. was
supported in part by a PBC fellowship of the Israel Council for Higher
Education. The work at NICPB was supported by the Estonian Research Council
Grant No. PRG736 an by the European Research Council (ERC) under Grant
Agreement No. 885413. M.K. was supported by NSF DMREF grant DMR-1629059.

##  Appendix A Estimation of heating in excitation laser spot

We use three methods to estimate the heating rate, a measure of the
temperature increase per unit laser power in the focused laser spot (K$/$mW):
(i) Stokes/anti-Stokes Raman scattering intensity ratio analysis; (ii)
monitoring laser power that is inducing the phase transition; and (iii) a
thermoconductivity model calculation. We use data for Ta2NiSe5 as an example
to illustrate these methods.

For the first method, we note that the Stokes scattering cross section $I_{S}$
and Anti-Stokes cross section $I_{AS}$ are related by the detailed balance
principle [22]

$nI_{S}(\omega)=(n+1)I_{AS}(\omega)~{},$ (18)

in which $n$ stands for the Bose factor

$n(\omega,T)=\frac{1}{exp(\hbar\omega/k_{B}T)-1}~{},$ (19)

where $\hbar$ is the reduced Planck’s constant, $\omega$ is frequency, $k_{B}$
is the Boltzmann’s constant, and $T$ is the temperature in the laser spot.

Using Eq. (18-19), the temperature can be derived as

$T=\frac{\hbar\omega}{k_{B}\ln(I_{S}/I_{AS})}~{},$ (20)

or, numerically,

$T[K]=\frac{11.605\;\omega}{\ln(I_{S}/I_{AS})}[meV]~{}.$ (21)

We studied the Stokes and Anti-Stokes cross section relation for the $ac$
scattering geometry at 295 K environmental temperature with various laser
power. In Fig. 25 we show the results measured with 8 mW laser power as an
example. The Raman responses calculated from Stokes and Anti-Stokes cross
sections match well with the temperature at the laser spot being 307 K. We use
Eq. (21) with the phonon intensity integrated from 7.5 to 8.5 meV to calculate
the temperature at the laser spot. By a linear fit to the power dependence of
the laser-spot temperature, we find the heating rate to be 1.29$\pm$0.17 K$/$
mW.

Figure 25:  Stokes-Anti-Stokes analysis of the $ac$ spectra for Ta2NiSe5. (a)
The Stokes and Anti-Stokes cross sections measured at environmental
temperature $T_{E}$ = 295 K with laser power P = 8 mW. (b) The Raman response
calculated from the measured spectra in (a). The laser-spot temperature, which
appears in the Bose factor $n$, is 307 K. The spectral resolution is 0.19 meV
for panels (a-b). (c) The laser-power dependence of the laser-spot
temperature. The straight line represents a linear fit.

For the second method, we gradually increase laser power at 295 K
environmental temperature. We find that when the laser power is in the range
of 26$\pm$4 mW, the domain stripes, which are visible across the sample
surface under laser illumination, disappear inside the laser spot. Moreover,
the temperature dependence of phonon width and intensity has a sudden change.
We assume that at this laser power, the temperature at the laser spot reaches
the transition temperature 328 K [9, 8], yielding the heating rate of
1.25$\pm$0.20 K$/$ mW. We note that for Ta2NiS5, we do not observe domain
stripes below its transition temperature 120 K. Its transition temperature is
identified by the sudden change of phonon intensity, see Fig. 18(b) for
example.

Figure 26:  The geometry of the heated volume. The lengths $l_{a}$ and $l_{c}$
are determined by the spot size, while $l_{b}$ is approximated by the skin
depth; the $S_{b}$ is in the cleave plane facing the vacuum, while the other
faces are in the bulk. Within the region marked by dashed lines, the heat flow
is mostly along $b$ axis, while outside the region heat flows in all
directions comparably, such that the temperature increase above the base
temperature $\Delta T$ decays as $1/r$, $r$ being the distance from the heated
region.

For the third method, we consider a heated region shown in Fig. 26 of a
rectangular shape, with the dimensions given by the beam spot size and the
penetration depth. The static heat equation takes the form:

$\begin{gathered}-\kappa_{x}\frac{\partial^{2}T}{\partial
x^{2}}-\kappa_{y}\frac{\partial^{2}T}{\partial
y^{2}}-\kappa_{z}\frac{\partial^{2}T}{\partial z^{2}}=\mathcal{P}({\bf r}),\\\
\left.\frac{\partial T({\bf r})}{\partial z}\right|_{y=0}=0;\\\ \partial
T(x\to\pm\infty,y\to-\infty,z\to\pm\infty)=T_{0}.\end{gathered}$ (22)

where $\mathcal{P}({\bf r})$ is the power density. We ignore the cooling
provided by the helium gas flow above the sample surface, so that there is no
heat flow at the $y=0$ boundary, while deep inside the bulk the temperature
should reach the base temperature $T_{0}$. Equation (22) can be solved using
Green’s function of the Laplace equation and image method to satisfy the first
boundary condition. In the latter, we extend the equation formally to $y>0$
half-space and add an ”image” power density

$\mathcal{P}(x,y,z)\theta(-y)\to\mathcal{P}(x,y,z)\theta(-y)+\mathcal{P}(x,-y,z)\theta(y).$
(23)

The solution is then given by:

$T({\bf
r})-T_{0}=\int\frac{d\tilde{x}^{\prime}d\tilde{y}^{\prime}d\tilde{z}^{\prime}}{4\pi}\frac{\mathcal{P}(\sqrt{\kappa_{x}}\tilde{x}^{\prime},...)\theta(-\tilde{y}^{\prime})+\tilde{y}^{\prime}\to-\tilde{y}^{\prime}}{|\tilde{\bf{r}}-{\bf\tilde{r}}^{\prime}|},$
(24)

where
$\tilde{\bf{r}}=(x/\sqrt{\kappa_{x}},y/\sqrt{\kappa_{y}},z/\sqrt{\kappa_{z}})$.
The solution decays as $1/r$ at large distances and thus satisfies the second
boundary condition of Eq. (22). An analytical result can be obtained for

$\mathcal{P}(x,y,z)=\frac{tP}{l_{a}l_{b}l_{c}}\theta(l_{a}/2-|x|)\theta(l_{b}-|y|)\theta(l_{c}/2-|z|)$
(25)

at ${\bf r}=0$ in the limit $l_{b}^{2}/\kappa_{b}\ll
l_{a,c}^{2}/\kappa_{a,c}$, where $P$ is the laser power and $t$ is the
transmission coefficient of the sample. Performing the integral one obtains

$T(0)-T_{0}\approx\frac{tP\left(\frac{l_{a}}{\sqrt{\kappa_{a}}}\ln\frac{\frac{l_{c}}{\sqrt{\kappa_{c}}}+\sqrt{\frac{l_{c}^{2}}{\kappa_{c}}+\frac{l_{a}^{2}}{\kappa_{a}}}}{l_{a}/\sqrt{\kappa_{a}}}+a\leftrightarrow
c\right)}{\sqrt{\kappa_{b}}\pi l_{a}l_{c}}.$ (26)

Qualitatively, the result can be understood as follows. Due to the anisotropic
shape of the heated region, right below it most heat is transferred along the
$b$ axis. However, the one-dimensional heat equation would result in a linear
solution $T(y)$, depending on the cutoff scale $l_{c}^{eff}$. It can be
estimated from the condition of the heat flow to the lateral direction being
equal to the heat flow along $b$. Approximating the temperature gradient along
$a,b,c$ as $\Delta T/l_{a,b,c}$ one gets the condition

$2l_{b}^{eff}l_{a}\kappa_{c}\frac{\Delta
T}{l_{c}}+2l_{b}^{eff}l_{c}\kappa_{a}\frac{\Delta
T}{l_{a}}=l_{a}l_{c}\kappa_{b}\frac{\Delta T}{l_{b}^{eff}},$ (27)

that results in the estimate

$l_{b}^{eff}=\sqrt{\frac{l_{a}^{2}l_{c}^{2}\kappa_{b}}{2l_{a}^{2}\kappa_{c}+2l_{c}^{2}\kappa_{a}}}.$
(28)

Equating the total heat flow outside this region to the input power one
obtains the estimate

$\Delta
T\approx\frac{tP}{\sqrt{8\kappa_{b}(l_{a}^{2}\kappa_{c}+l_{c}^{2}\kappa_{a})}}.$
(29)

Overall, one notices that the temperature increase scales with the linear size
of the spot. The laser heating power $R$ is then determined as $\Delta T/P$;
for the actual estimate we thus need to find the transmission coefficient $t$
first.

The transmission coefficient can be calculated from the complex index of
refraction $n$ by the relationship $t=1-|(n-1)/(n+1)|^{2}$:

$t=\frac{4n_{1}}{(n_{1}+1)^{2}+n_{2}^{2}}~{},$ (30)

in which $n_{1}$ and $n_{2}$ are the real and imaginary part of $n$. The
complex index of refraction can be obtained from the complex dielectric
constant $\epsilon$ by the relationship $\epsilon=n^{2}$:

$n_{1}=\sqrt{\frac{\sqrt{\epsilon_{1}^{2}+\epsilon_{2}^{2}}+\epsilon_{1}}{2}}~{},n_{2}=\sqrt{\frac{\sqrt{\epsilon_{1}^{2}+\epsilon_{2}^{2}}-\epsilon_{1}}{2}}~{},$
(31)

where $\epsilon_{1}$ and $\epsilon_{2}$ are the real and imaginary part of
$\epsilon$. The imaginary part $\epsilon_{2}$ can be calculated from the real
part of the optical conductivity $\sigma_{1}$:

$\epsilon_{2}=\frac{60}{\omega}\sigma_{1}~{},$ (32)

in which the unit of $\omega$ is cm-1 and that of $\sigma_{1}$ is
$\Omega^{-1}$cm-1. The polarization of the incoming light is along a-axis of
the sample. Because the quantities $\epsilon_{1}$ and $\sigma_{1}$ are 8.13
and 2.47 $\Omega^{-1}$cm-1 respectively at the laser wavelength [55], we find
t=0.64.

The experimentally measured thermal conductivity [60] at 295 K is $\kappa_{a}$
= 208 mWK-1cm-1 and $\kappa_{c}$ = 57.4 mWK-1cm-1.

The length scales $l_{a}=5\times 10^{-3}$ cm and $l_{c}=1\times 10^{-2}$ cm is
determined by the $50\times 100$ $\mu$m2 laser spot. The length $l_{b}$ is the
skin depth, which is calculated from the imaginary part of the index of
refraction $n_{2}$:

$l_{b}=\frac{1}{2\pi\omega n_{2}}~{},$ (33)

in which the unit of $\omega$ is cm-1 and that of $l_{b}$ is cm. At the laser
wavelength, $n_{2}$=1.5, and we find $l_{b}=7\times 10^{-6}$ cm (70 nm).

The known values for the variables in Eq. (26) are summarized in Table 5.
Because $\kappa_{b}$ is unknown, we cannot use Eq. (26) to calculate the laser
heating rate. However, if the heating rate is 1.29 K$/$ mW at 295 K, as
determined from the Stokes/anti-Stokes analysis, the value for $\kappa_{b}$
should be 1.38 mWK-1cm-1.

Table 5: The various parameters used in calculating the laser heating rate at 295 K. The transmission coefficient is 0.64. Quantity (unit) | a-axis | c-axis
---|---|---
$\kappa$ (mWK-1cm-1) | 208 | 57.4
$l$ (10-3cm) | 5 | 10

An estimate for the electronic contribution to thermal conductivity
$\kappa^{(e)}$ could be calculated from the electric resistivity $\rho$ by
virtue of Wiedemann-Franz law:

$\kappa^{(e)}=\frac{LT}{\rho}~{},$ (34)

in which $L=2.44\times 10^{-5}$ mW$\Omega$K-2 is the Lorenz number. The
resistivity along b axis at 295 K is $\rho_{b}=2.34\times 10^{-1}$ $\Omega$cm
[61], and we find $\kappa^{(e)}_{b}$ = 0.0308 mWK-1cm-1. The ratio of
$\kappa^{(e)}_{b}$ to $\kappa_{b}$ is consistent with the ratios obtained
along a and c axes [60]. Moreover, that $\kappa_{b}\ll\kappa_{a,c}$ is
consistent with the quasi-2D structure of the system [62].

To illustrate that the laser heating rate is strongly temperature dependent,
we use Eq. (26) to calculate the heating rate as a function of temperature.
The temperature dependence of $\kappa_{a}$ and $\kappa_{c}$ are taken from
Ref. [60]. We assume that the ratio of $\kappa_{b}$ to $\kappa_{c}$ is
temperature independent, and fix this ratio to be $1.38/57.4=2.40\%$. In Fig.
27 we show the calculated heating rate. Because of such assumption, the laser
heating rate at 295 K is the same as that determined from the Stokes/anti-
Stokes analysis.

Figure 27: Temperature dependence of the laser heating rate R, calculated by
Eq. (26) with the thermal conductivity data taken from Ref. [60].

##  Appendix B The fitting model for the Fano interference

Above $T_{c}$ , the B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$ phonon modes of
Ta2NiSe5 exhibit strongly asymmetric Fano lineshapes. To analyze the physics
of this Fano interference features, we propose a model describing three phonon
modes individually coupled to broad excitonic continuum. The total Raman
response is described in the following way:

$\chi^{\prime\prime}\sim\Im T^{T}GT~{},$ (35)

where $T^{T}=(\begin{array}[]{cc}T_{ph}^{T}&t_{e}\end{array})$ denotes the
vertices for light scattering process (the subscript ”T” denotes ”Transpose”),
in which $T_{ph}^{T}=(\begin{array}[]{ccc}t_{p1}&t_{p2}&t_{p3}\end{array})$.
$G$ comprises the Green’s functions for the interacting phononic and excitonic
excitations that can be obtained by solving the Dyson equation:

$G=(G_{0}^{-1}-V)^{-1}.$ (36)

In Eq. (36)

$G_{0}=\begin{pmatrix}G_{ph}^{0}&0\\\ 0&G_{e}^{0}\end{pmatrix}~{}$ (37)

is the bare Green’s function and

$V=\begin{pmatrix}0&V_{e-ph}\\\ V_{e-ph}^{T}&0\end{pmatrix}~{}$ (38)

is the exciton-phonon interaction. Here the bare phononic Green’s function

$G_{ph}^{0}=\operatorname{diag}[G_{pi}^{0}]\\\
=\operatorname{diag}[-(\frac{1}{\omega-\omega_{pi}+i\gamma_{pi}}-\frac{1}{\omega+\omega_{pi}+i\gamma_{pi}})]~{},$
(39)

correspond to the B${}_{2g}^{(i)}$-symmetry phonon modes (i=1,2,3),

$G_{e}=\frac{1}{\omega_{e}^{2}/\gamma_{e}-i\omega}~{}$ (40)

represents the excitonic continuum, and
$V_{e-ph}^{T}=(\begin{array}[]{ccc}v_{1}&v_{2}&v_{3}\end{array})$ denotes
exciton interaction strength with the corresponding phonons. For phononic
equation, the parameters $\omega_{pi}$ and $\gamma_{pi}$ have the meaning of
the mode frequency and the half width at half maximum (HWHM), respectively.
For the excitonic equation, $\omega_{e}$ is the energy of the overdamped
excitations and $\gamma_{e}$ represents the relaxation rate. The quantity
$\Omega_{e}=\omega_{e}^{2}/\gamma_{e}$ denotes the energy at which the
excitonic continuum has maximum intensity.

The bare phononic and excitonic responses are

$\chi^{\prime\prime(0)}_{p}=\Im
T_{ph}^{T}G_{ph}^{0}T_{ph}=\sum_{i=1}^{3}t_{pi}^{2}\Im G_{pi}^{0}\,$ (41)

and

$\chi^{\prime\prime(0)}_{e}=t_{e}^{2}\Im G_{e}^{0}\,.$ (42)

These two expressions are related to Eqsw. (1-2) in the main text.

The renormalized phononic and excitonic responses, $\chi^{\prime\prime}_{p}$
and $\chi^{\prime\prime}_{e}$, as well as the interference term
$\chi^{\prime\prime}_{int}$ are calculated from the renormalized Green’s
function $G$:

$G=\begin{pmatrix}G_{ph}&G_{e-ph}\\\ G_{e-ph}^{T}&G_{e}\end{pmatrix}~{},$ (43)

where $G_{e-ph}^{T}=(\begin{array}[]{ccc}G_{1e}&G_{2e}&G_{3e}\end{array})$
represents the interference between the phononic modes and the continuum.

The phononic term is calculated from the 3x3 phononic block:

$\chi^{\prime\prime}_{p}=\Im T_{ph}^{T}G_{ph}T_{ph}\,;$ (44)

the excitonic term results from the 1x1 excitonic block:

$\chi^{\prime\prime}_{e}=t_{e}^{2}\Im G_{e}\,;$ (45)

and the interference term is given by the off-diagonal block

$\chi^{\prime\prime}_{int}=t_{e}\sum_{i=1}^{3}t_{pi}\Im G_{ie}\,.$ (46)

These three terms, as defined, satisfy Eq. (4):

$\chi^{\prime\prime}=\chi^{\prime\prime}_{p}+\chi^{\prime\prime}_{e}+\chi^{\prime\prime}_{int}.$
(47)

For the $ac$ spectra below $T_{c}$ , especially at about 300 K, this model
cannot properly account for the low frequency Raman response. This additional
spectral feature is related to Raman coupling to the longitudinal acoustic
excitations at finite momenta in the presence of quasi-periodic domain walls,
detailed in the main text. The Green’s function of each individual acoustic
mode has form similar to the phononic Green’s function:

$G_{s}(\omega,q)=-(\frac{1}{\omega-
c_{s}q+ir_{s}q}-\frac{1}{\omega+c_{s}q+ir_{s}q})~{}.$ (48)

Its light-scattering vertex $t_{s}(q)$ and coupling to the excitonic continuum
$v_{s}(q)$ are both proportional to the square root of wavevector:
$t_{s}=\tau_{s}\sqrt{q}$ and $v_{s}=\beta_{s}\sqrt{q}$. The frequency
$\omega_{s}$ and the HWHM $\gamma_{s}$ are both proportional to the
wavevector: $\omega_{s}=c_{s}q$ and $\gamma_{s}=r_{s}q$.

After introducing this mode, Eqs. (37-38) extend to

$G_{0}^{\prime}=\begin{pmatrix}G_{ph}^{0}&0&0\\\ 0&G_{s}&0\\\
0&0&G_{e}\end{pmatrix}\,,$ (49) $V^{\prime}=\begin{pmatrix}0&0&V_{e-ph}\\\
0&0&\beta_{s}\sqrt{q}\\\ V_{e-ph}^{T}&{\beta_{s}}\sqrt{q}&0\end{pmatrix}\,,$
(50)

and the vertex for light-scattering process becomes
$T^{{}^{\prime}T}=\left(\begin{array}[]{ccc}T_{ph}^{T}&\tau_{s}\sqrt{q}&t_{e}\end{array}\right)$.

Then the total response function containing coupling to acoustic mode for
domain walls $i$ lattice constants apart is given by

$\chi^{\prime\prime}_{i}\sim\Im T^{{}^{\prime}T}G^{\prime}T^{\prime}~{}.$ (51)

Due to the requirement of causality, the real part of susceptibility should be
an even function while the imaginary part should be an odd function.
Therefore, the static Raman susceptibility is purely real. The total static
Raman susceptibility is given by

$\chi(\omega=0)\sim\Re T^{T}G(\omega=0)T~{}.$ (52)

The bare phononic static Raman susceptibility can be obtained from

$\chi_{p}^{(0)}(\omega=0)\sim\Re
T_{ph}^{T}G_{ph}^{0}(\omega=0)T_{ph}=\sum_{i=1}^{3}\frac{2t_{pi}^{2}\omega_{pi}}{\omega_{pi}^{2}+\gamma_{pi}^{2}}~{},$
(53)

Similarly, the bare excitonic static Raman susceptibility is given by

$\chi_{e}^{(0)}(\omega=0)\sim t_{e}^{2}\Re
G_{e}^{0}(\omega=0)=\frac{t_{e}^{2}}{\Omega_{e}}~{},$ (54)

For temperature-independent $t_{e}$, the temperature dependence of
$1/\chi_{e}$ directly follows that of
$\Omega_{e}(T)=\omega_{e}^{2}/\gamma_{e}$.

Using the Green’s function $G^{\prime}$, which describes the interacting
phononic (both optical and acoustic) and excitonic excitations, we can
evaluate the renormalization of sound velocity above $T_{c}$ . Because the
relevant frequencies for the acoustic mode are much lower than that of the
optical or excitonic modes, we can keep $\omega$ only in the bare acoustic
mode’s Green’s function:

$G^{*}=\begin{pmatrix}\frac{\omega_{p1}}{2}&0&0&0&-v_{1}\\\
0&\frac{\omega_{p2}}{2}&0&0&-v_{2}\\\ 0&0&\frac{\omega_{p3}}{2}&0&-v_{3}\\\
0&0&0&-\frac{\omega^{2}-c_{s}^{2}q^{2}}{2c_{s}q}&-\beta_{s}\sqrt{q}\\\
-v_{1}&-v_{2}&-v_{3}&-\beta_{s}\sqrt{q}&\Omega_{e}\end{pmatrix}^{-1}~{},$ (55)

in which finite linewidths of phonon modes are neglected. The pole of this
Green’s function is obtained by requiring the determinant of inverse $G^{*}$
to be zero:

$\frac{\omega^{2}-c_{s}^{2}q^{2}}{2c_{s}q}=-\frac{\beta_{s}^{2}q}{\Omega_{e}-\sum_{i=1}^{3}\frac{2v_{i}^{2}}{\omega_{pi}}}.$
(56)

Identifying $\omega^{2}$ as $\tilde{c}_{s}^{2}q^{2}$, in which $\tilde{c}_{s}$
represents the renormalized sound velocity above $T_{c}$ , we have

$\tilde{c}_{s}^{2}=c_{s}^{2}-\frac{2\beta_{s}^{2}c_{s}}{\Omega_{e}-\sum_{i=1}^{3}\frac{2v_{i}^{2}}{\omega_{pi}}}.$
(57)

##  Appendix C Illustration of the fitting model for the Fano interference

Figure 28:  Coupling of the excitonic continuum with (a) the B${}_{2g}^{(1)}$
phonon mode and (b) the B${}_{2g}^{(2)}$ phonon mode. The parameters used for
plotting are obtained from fitting the 380 K spectrum of Ta2NiSe5. The Raman
response $\chi^{\prime\prime}(\omega)$ of the excitonic and phononic
excitations for cases without (dashed lines) and with (solid lines) the effect
of coupling are compared. The solid red lines correspond to the addition of
the solid blue, solid green, and solid orange lines.

To provide more insights into the fitting model described in Appendix B, we
consider a simplified case in which only one phonon mode couples to an
excitonic continuum. When exciton-phonon interaction $v$ is zero, the total
Raman response is reduced to the sum of the excitonic component and the
phononic component:

$\chi^{\prime\prime}_{0}(\omega)=\frac{t_{e}^{2}\omega}{\Omega_{e}^{2}+\omega^{2}}+\frac{4t_{p}^{2}\gamma_{p}\omega_{p}\omega}{(\omega^{2}-\omega_{p}^{2})^{2}+2\gamma_{p}^{2}(\omega^{2}+\omega_{p}^{2})+\gamma_{p}^{4}}~{},$
(58)

in which $\Omega_{e}$ stands for $\omega_{e}^{2}/\gamma_{e}$.

In Fig. 28 we show the excitonic continuum coupling with the B${}_{2g}^{(1)}$
and B${}_{2g}^{(2)}$ phonon modes in an individually way to illustrate the
effect of coupling. First we discuss the renormalization effect. The
renormalized phonon mode shifts in energy and broadens in lineshape. If we
neglect the bare phonon width $\gamma_{p}$ for simplicity, the central energy
of the renormalized phonon mode, $\omega_{pv}$, can be expressed as

$\omega_{pv}=\omega_{p}+\frac{v^{2}(v^{2}-2\omega_{p}\Omega_{e})}{2(v^{2}\Omega_{e}+\omega_{p}\Omega_{e}^{2}+\omega_{p}^{3})}~{}.$
(59)

For both the B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$ phonon modes, $v^{2}$ is
smaller than $2\omega_{p}\Omega_{e}$; therefore $\omega_{pv}$ is smaller than
$\omega_{p}$, meaning the renormalized phonon mode shifts to lower energy.
Interestingly, at frequency $\omega_{pv}$ the renormalized excitonic continuum
has the same intensity as the unrenormalized continuum; below $\omega_{pv}$
the excitonic response is enhanced while above $\omega_{pv}$ the excitonic
response is suppressed.

Figure 29:  The influence of the coupling strength $v$, and the frequency at
which the excitonic continuum has maximum intensity
$\Omega_{e}=\omega_{e}^{2}/\gamma_{e}$ on the Raman response
$\chi^{\prime\prime}(\omega)$ of the coupled excitonic and phononic modes. (a)
The combined response for varying $v/\omega_{p}$ with fixed
$\Omega_{e}/\omega_{p}$=1.3; (b) The combined response for varying
$\Omega_{e}/\omega_{p}$ with fixed $v/\omega_{p}$=0.3. For all panels,
$t_{p}$=3.8 arb. units; $t_{e}$=7.2 arb. units; $\omega_{p}$=7.8 meV;
$\gamma_{p}$=0.38 meV. The ratios derived from the 380 K spectrum of Ta2NiSe5
are $\Omega_{e}/\omega_{p}$=1.3 and $v/\omega_{p}$=0.3. The horizontal axis is
normalized to the phonon frequency $\omega_{p}$.

Second we discuss the interference effect. For the interference term,
corresponding to the off-diagonal elements of the renormalized Green’s
function, there is a frequency $\omega_{int}$ at which it changes the sign.
The frequency $\omega_{int}$ has the following expression

$\omega_{int}=\sqrt{2\Omega_{e}\gamma_{p}+\gamma_{p}^{2}+\omega_{p}^{2}}.$
(60)

We note that $\omega_{int}$ is a bit larger than $\omega_{p}$. The shape of
the interference term is controlled by sign of $v$: for positive $v$, below
the zero-intensity point the interference term has positive intensity while
the intensity is negative above the zero-intensity point; for negative $v$,
the opposite is true. Far away from the resonance, the interference term
decays to zero because the phonon mode has negligible intensity at such high
energy.

Finally we note that when $\omega$ is much larger than $\omega_{p}$ and
$\gamma_{e}$, the Raman response is In Fig. 29 we show how the coupling
strength $v$, and the energy at which the excitonic continuum has maximum
intensity $\Omega_{e}=\omega_{e}^{2}/\gamma_{e}$ influence the combined Raman
response. The lineshape becomes more asymmetric with increasing $v/\omega_{p}$
[Fig. 29 (a)]. When $\Omega_{e}$ is varied, we find that the lineshape becomes
more asymmetric for smaller $\Omega_{e}$ [Fig. 29 (b)].

approximately $t_{e}^{2}/\omega$. Therefore, $t_{e}$ can be determined by
fitting the data at large $\omega$.

##  Appendix D The fitted parameters of the anharmonic decay model for
Ta2NiSe5 and Ta2NiS5

The temperature dependence of both frequency and FWHM for the most phonon
modes of Ta2NiSe5 and Ta2NiS5 modes can be accounted by the anharmonic decay
model [Eq. (16-17)]. The values for the parameters from fitting are summarized
in Table 6.

Table 6:  The fitting parameters of the anharmonic decay model [Eq. (16-17)] for the Raman-active optical phonon modes of Ta2Ni(Se1-xSx)5 family. The frequencies and FWHM below the phase transition temperature are fitted. The anharmonic decay model is not applicable for the B2g-symmetry modes of Ta2NiS5. The units are in meV. Mode | $\omega_{0}$ | $\omega_{2}$ | $\Gamma_{0}$ | $\Gamma_{2}$
---|---|---|---|---
Ta2NiSe5
A${}_{g}^{(1)}$ | 4.235(16) | 0.0038(10) | 0.06(4) | 0.0014(15)
B${}_{2g}^{(1)}$ | 8.721(5) | 0.0290(11) | 0.073(15) | 0.019(3)
A${}_{g}^{(2)}$ | 12.484(12) | 0.0405(22) | 0.09(3) | 0.025(6)
B${}_{2g}^{(2)}$ | 15.35(4) | 0.070(15) | 0.05(10) | 0.09(4)
A${}_{g}^{(3)}$ | 17.077(12) | 0.191(5) | 0.08(5) | 0.114(23)
B${}_{2g}^{(3)}$ | 18.45(3) | 0.050(16) | 0.11(10) | 0.08(5)
A${}_{g}^{(4)}$ | 22.28(3) | 0.082(15) | 0.12(8) | 0.05(4)
A${}_{g}^{(5)}$ | 24.287(11) | 0.129(6) | 0.05(3) | 0.072(15)
A${}_{g}^{(6)}$ | 27.17(4) | 0.137(16) | 0.23(11) | 0.09(5)
A${}_{g}^{(7)}$ | 29.62(6) | 0.21(3) | 0.18(16) | 0.12(9)
A${}_{g}^{(8)}$ | 36.75(14) | 0.30(8) | 0.3(4) | 0.09(24)
Ta2Ni(Se0.75S0.25)5
A${}_{g}^{(1)}$ | 4.277(16) | 0.0039(9) | 0.12(5) | 0.0025(34)
B${}_{2g}^{(1)}$ | 8.758(7) | 0.0322(15) | 0.252(21) | 0.028(4)
Ta2Ni(Se0.33S0.67)5
A${}_{g}^{(1)}$ | 4.630(12) | 0.0048(8) | 0.10(2) | 0.0012(11)
B${}_{2g}^{(1)}$ | 8.739(19) | 0.032(4) | 0.479(26) | 0.022(4)
Ta2NiS5
A${}_{g}^{(1)}$ | 5.025(10) | 0.0040(7) | 0.068(25) | 0.0014(19)
A${}_{g}^{(2)}$ | 16.198(4) | 0.0610(13) | 0.051(9) | 0.0359(26)
A${}_{g}^{(3)}$ | 19.276(11) | 0.134(4) | 0.11(3) | 0.070(11)
A${}_{g}^{(4)}$ | 33.92(4) | 0.111(19) | 0.28(11) | 0.15(6)
A${}_{g}^{(5)}$ | 36.634(10) | 0.138(5) | 0.056(28) | 0.112(16)
A${}_{g}^{(6)}$ | 40.29(20) | 0.21(12) | 0.6(7) | 0.3(4)
A${}_{g}^{(7)}$ | 43.39(6) | 0.29(4) | 0.01(14) | 0.17(9)
A${}_{g}^{(8)}$ | 49.72(3) | 0.23(3) | 0.04(10) | 0.45(8)

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