4                    $10^{-3}$                       5                       1                  1         63.8            $10^{-4}$                        5                False             True          63.1

4                    $10^{-3}$                       5                       1                  1         63.8            $10^{-4}$                        1                False             True          63.4

5                    $10^{-3}$                       5                       1                  1         63.8            $10^{-4}$                        5                False            False          63.5

6                    $10^{-3}$                       5                      10                  1         63.8            $10^{-4}$                        5                False            False          63.6

7                    $10^{-4}$                       5                       1                  1         63.8            $10^{-4}$                        5                False            False          63.6

8                    $10^{-3}$                       5                       1                  1         63.8            $10^{-4}$                        5                False            False          63.6

9                    $10^{-4}$                       5                       1                  1         63.9            $10^{-4}$                        5                False            False          63.6

10                   $10^{-4}$                       5                       1                  1         64.2            $10^{-4}$                        5                False            False          63.7

11                   $10^{-4}$                       5                       5                  1         64.3            $10^{-4}$                        5                False            False          63.8

12                   $10^{-4}$                       5                       5                  1         64.5            $10^{-4}$                        5                False            False          63.9

13                   $10^{-4}$                       5                       5                  1         64.6            $10^{-4}$                        5                False            False          63.9

14                   $10^{-4}$                       5                       1                  2         64.7            $10^{-4}$                        5                False            False          63.8

15                   $10^{-4}$                       5                       1                  1         64.8            $10^{-4}$                        5                False            False          63.4

16                   $10^{-4}$                       5                       1                  1         64.8            $10^{-3}$                       10                False            False          63.2

17                   $10^{-4}$                       5                       1                  1         64.8            $10^{-4}$                       10                False            False          63.3

18                   $10^{-4}$                       5                      10                  1         64.8            $10^{-4}$                       10                False            False          63.5

19                   $10^{-4}$                       5                       5                  1         64.8            $10^{-4}$                       10                False            False          63.6

20                   $10^{-4}$                       5                       5                  1         64.7            $10^{-4}$                       10                False            False          63.8

21                   $10^{-4}$                       5                      10                  1         64.7            $10^{-4}$                       10                False            False          63.9

21                   $10^{-4}$                       5                      10                  1         64.7            $10^{-4}$                       10                False            False          63.8

22                   $10^{-4}$                       5                       5                  1         64.7            $10^{-4}$                       10                False            False          63.9

23                   $10^{-4}$                       5                      10                  1         64.7            $10^{-4}$                       10                False            False          63.8

24                   $10^{-4}$                       5                      10                  1         64.7               —                        —                  —              —           —

Atari, per-game performance across $3$ seeds. Shading depicts standard deviation.

Mean per-game performance between 190-200M frames.

KL      KL & V      KL & V, L=4        KL-S          L2       STACX           V

Alien                   45677       44880            58067       35750       50692       31809        7964

Amidar                   4800        7099             7528        4974        7691        3719        1896

Assault                 20334       29473            33019       21747       28301       19648        4101

Asterix                511550      439475           533385      487367        6798      245617       86053

Asteroids              145337      238320           289689        8585      220366      156096       56577

Atlantis               831920      813772           814780      806698      854441      848007      648988

Bank Heist                571        1325                0          13        1165        1329        1339

Battle Zone             73323       88407            88350       78941       50453       78359       72787

Beam Rider              37170       51649            57409       41454       67726       62892       74397

Berzerk                 21146        2946             1588        2183         240        1523        1069

Bowling                    46          50               42          46          50          28          52

Boxing                    100         100               86         100         100         100         100

Breakout                  742         832              847         774         827         717          16

Centipede              537032      542730           558849      291569      550394      478347        8895

Chopper Command        830772      934863           838090      736012       11274      846788      341350

Crazy Climber          233445      212229           265729      199150      229496      182617      126353

Defender               393457      374012           421894      364053       69193      344453       55152

Demon Attack           132508      133109           133571      132529      133469      130741      129863

Double Dunk                22          23               23          21          23          24          23

Enduro                   2349        2349             2350        2360        2365         259        2187

Fishing Derby              41          63               68          41          52          62          59

Freeway                    10          30               25          31          30          18          33

Frostbite                8820        3895             3995        5547       13477        2522        1669

Gopher                 116010      116037           122459       92185      122790       87094       11920

Gravitar                  271         709              748         259        3594        2746         944

Hero                    60896       48551            52432       56044       51631       35559       20235

Ice Hockey                  5          15               20           4          15          19          20

Jamesbond               22129       25951            30157       25766       18200       26123       23263

Kangaroo                12200       12557            13174        1940       13235        3182        8722

Krull                   10750        9768            10510       11156       10502       10480        8899

Kung Fu Master          51038       58732            54354       54559       63632       67823       54584

Montezuma Revenge           0           0                0           0           0           0           0

Ms Pacman               25926       22876            28279       26267       27564       12647        2759

Name This Game          31203       31863            36838       30912       32344       24616       12583

Phoenix                529404      542998           658082      407520      440821      370270      247854

Pitfall                     0          -1                0           0           0           0           0

Pong                       21          21               21          21          21          21          21

Private Eye               165         144               98         130          67         100          68

Qbert                   87214       37135            72320       30047       75197       27264        3901

Riverraid              129515      132751            32300       91267      177127       47671       26418

Road Runner            240377       61710           521596       17002      424588       62191       34773

Robotank                   64          66               71          65          64          61          65

Seaquest               684870        2189            82925      616738        1477        1744        3653

Skiing                 -10023       -8988            -9797       -8988       -9893      -10504      -13312

Solaris                  2120        2182             2188        1858        2194        2326        2202

Space Invaders          35762       54046            40790       11314       49333       34875       15424

Star Gunner            588377      663477           790833      587411       39510      298448       43561

Surround                    9           9               10           9           9           3           9

Tennis                     23          24               23          21          24          19          24

Time Pilot              94746       60918            68626       95854       93466       49932       40127

Tutankham                 282         268              291         280         288         101         205

Up N Down              342121      303741           381780      109392      202715      315588       17252

Venture                     0           0                0           0           0           0           0

Video Pinball          230252      479861           399094      505212      485852      441220       77100

Wizard Of Wor           21597       45731            49806       22936       10817       47854       24250

Yars Revenge            77001      286734           408061       32031      398656      113651       77169

Zaxxon                  44280       49448            59011       36261       49734       56952       35494


Present address: ]CEA-Leti, Université Grenoble Alpes, Grenoble, France
Present address: ]4th Physical Institute – Solids and Nanostructures,
University of Göttingen, 37077 Göttingen, Germany

# Anomalous phonon Grüneisen parameters in semiconductor Ta2NiS5

Mai Ye<EMAIL_ADDRESS>Institute for Quantum Materials and Technologies,
Karlsruhe Institute of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany
Tom Lacmann  Institute for Quantum Materials and Technologies, Karlsruhe
Institute of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany  Mehdi
Frachet  [  Institute for Quantum Materials and Technologies, Karlsruhe
Institute of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany  Igor
Vinograd  [  Institute for Quantum Materials and Technologies, Karlsruhe
Institute of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany  Gaston
Garbarino  European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble
Cedex, France  Nour Maraytta  Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, Kaiserstr. 12, 76131
Karlsruhe, Germany  Michael Merz  Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, Kaiserstr. 12, 76131
Karlsruhe, Germany  Karlsruhe Nano Micro Facility (KNMFi), Karlsruhe Institute
of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany  Rolf Heid  Institute
for Quantum Materials and Technologies, Karlsruhe Institute of Technology,
Kaiserstr. 12, 76131 Karlsruhe, Germany  Amir-Abbas Haghighirad  Institute for
Quantum Materials and Technologies, Karlsruhe Institute of Technology,
Kaiserstr. 12, 76131 Karlsruhe, Germany  Matthieu Le Tacon
<EMAIL_ADDRESS>Institute for Quantum Materials and Technologies,
Karlsruhe Institute of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany

###### Abstract

Strain tuning is a powerful experimental method in probing correlated electron
systems. Here we study the strain response of the lattice dynamics and
electronic structure in semiconductor Ta2NiS5 by polarization-resolved Raman
spectroscopy. We observe an increase of the size of the direct semiconducting
band gap. Although the majority of the optical phonons show only marginal
dependence to applied strain, the frequency of the two B2g phonon modes, which
have quadrupolar symmetry and already anomalously soften on cooling under zero
strain, increases significantly with tensile strain along the $a$ axis. The
corresponding Grüneisen parameters are unusually large in magnitude and
negative in sign. These effects are well captured by first-principles density
functional theory calculations and indicate close proximity of Ta2NiS5 to a
structural instability, similar to that encountered in excitonic insulator
candidate Ta2NiSe5.

##  I Introduction

A macroscopic number of bound pairs of fermions, formed at low temperatures
due to attractive interaction, result in quantum phases with superconductivity
[1] or superfluidity [2]. Excitonic insulators (EIs) belong to such quantum
phases, in that the Coulomb attraction in a semimetal or in a narrow-gap
semiconductor leads to an instability towards the formation of electron-hole
pairs (excitons) [3, 4, 5] and these bosonic quasiparticles can collectively
establish a coherent state and open an interaction-induced gap [6].
Furthermore, if the participating electron and hole states belong to bands of
different symmetry, the resulting EI breaks the crystal lattice symmetry.

The material family Ta2Ni(Se1-xSx)5 serves as a suitable platform to
investigate the rich physics of EIs. On one end of its phase diagram, Ta2NiSe5
is largely believed to be a semimetal at high temperature [7, 8] which has
been proposed to host an EI state [9] below an orthorhombic-to-monoclinic
structural phase transition around 326 K [10, 11]. This transition reduces the
point-group symmetry from $D_{2h}$ to $C_{2h}$ by breaking two mirror
symmetries, and by group theory the corresponding order parameter is of B2g
symmetry. It is accompanied with the opening of a gap in the electronic
dispersion and a band flattening [9], but there is generally a controversy on
whether these features can be effectively attributed to the condensation of
excitons [12, 13, 14, 15, 16], or they are only a consequence of the lattice
symmetry lowering and electron-phonon coupling [7, 17, 18].

The transition temperature is suppressed by S substitution [11, 19, 20], and
at the other end of the phase diagram, the iso-structural compound Ta2NiS5 is
a semiconductor with a band gap exceeding the exciton binding energy,
henceforth forbidding the EI instability [10, 21, 22]. In contrast with
Ta2NiSe5, the properties of semiconducting Ta2NiS5 have been relatively little
explored. It is for instance unclear whether the aforementioned structural
transition is completely suppressed. This has been suggested in recent x-ray
diffraction experiments [22], but Raman scattering investigation [20] has
interpreted a temperature-dependent breaking of the phonon polarization
selection rules below 120 K as evidence for such transition. This view is
further supported by first-principles calculations [23] indicating that in
both Ta2NiSe5 and Ta2NiS5, the lattice symmetry is broken by phonon
instabilities, emphasizing the importance of lattice degrees of freedom in
shaping the phase diagram of this family of materials.

In this work, we tackle this issue from a different perspective, probing the
response of the phononic and electronic degrees of freedom of Ta2NiS5 to
uniaxial strain. Strain tuning has proven particularly insightful in studying
the electronic properties of unconventional low- [24, 25] or high-temperature
superconductors [26, 27, 28], ferromagnets [29], antiferromagnets [30], and
topological systems [31]. Here we combine it with Raman spectroscopy and
first-principles calculations to reveal the highly anomalous behavior of two
phonon modes of B2g symmetry involving shear vibrations of the Ta atoms along
the crystallographic a-direction. We show that this behavior arises
independently from changes of the band gap size and points towards an
intrinsic structural instability of the orthorhombic unit cell upon volume
reduction. More generally, our results demonstrate the possibility of
manipulating the electronic band gap in the Ta2Ni(Se1-xSx)5 family using
strain, which offers interesting perspectives to approach the still elusive EI
regime.

##  II Method

High-quality single crystals of Ta2NiS5 were grown using chemical vapor
transport [32]. The composition of the obtained crystals was performed using
energy dispersive x-ray spectroscopy (EDX) in a COXEM EM-30plus electron
microscope equipped with an Oxford Silicon-Drift-Detector (SDD) and
AZtecLiveLite-software package.

Strain-dependent Raman-scattering experiments were performed with a Horiba
Jobin-Yvon LabRAM HR evolution spectrometer. We used a He-Ne laser (632.8 nm)
with less than 1 mW power that was focused on the sample with a x50
magnification objective. The laser spot size was around 5 $\mu$m in diameter.
Spectra were recorded with 1800 mm-1 (0.6 cm-1 spectral resolution) and 600
mm-1 (1.6 cm-1 spectral resolution) gratings and a liquid-nitrogen-cooled CCD
detector. All spectra of Raman response were corrected for the instrumental
spectral response and Bose factor. Two polarization configurations were
employed to probe excitations in different symmetry channels, namely the $aa$
and $ac$ configurations in which the incident light polarization lies along
the crystallographic $a$ direction, and scattered one along $a$ or $c$
direction, respectively. According to Raman selection rules [33], in the
orthorhombic phase (point group D2h), the $aa$ and $ac$ geometries probe the
Ag and B2g Raman signal, respectively. Samples with a cleaved crystallographic
$ac$ plane were glued to Razorbill CS220T strain cell by Loctite Stycast
2850FT epoxy with CAT 24LV as the epoxy catalyst. The sample surfaces were
examined under polarized light to find an area without intrinsic strain. We
verified that the process of mounting the sample onto the strain cell does not
introduce additional stress by comparing the Raman spectra of unmounted and
mounted samples. The strain transmission was estimated from XRD measurement as
detailed in the supplemental material [32]; the corresponding data measured at
European Synchrotron Radiation Facility (ESRF) ID15B beamline are available
from the ESRF Data Portal [34].

Density functional theory (DFT) calculations were performed using the mixed-
basis pseudopotential method [35] to assess the effect of uniaxial strain on
the electronic structure and Raman active modes [32]. The exchange-correlation
functional was represented by the generalized gradient approximation (GGA)
[36]. The DFT + U approach was applied with U=10 eV for the Ni $d$ states. We
performed all calculations in the room-temperature orthorhombic structure
(space group $Cmcm$). Lattice constants were taken from Sunshine et al. [37],
$a=3.415$ Å, $b=12.146$ Å, $c=15.097$ Å. The relatively small tensile strain
was modeled by scaling $a$ only, ignoring any elastic reaction of the $b$ or
$c$ axes. For each structure, atomic positions were relaxed until forces were
smaller than 2.6 meV/Å. Raman-mode frequencies were calculated using the
linear response, or density-functional perturbation theory (DFPT), implemented
in the mixed-basis pseudopotential method [38].

##  III Results and Discussion

###  III.1 Strain-free phonon spectra

In agreement with previous Raman-scattering investigation [20], we observe a
"leakage" of certain phonon modes at low temperatures. This is illustrated by
the Raman spectra measured at 200 K and 15 K in two configurations, as shown
in Fig. 1. Up to 40 meV, and in agreement with the selection rules of the
orthorhombic phase, six Ag Raman phonons are observed in the $aa$ scattering
geometry and three B2g modes in the $ac$ scattering geometry. At low
temperature, on the contrary, the lowest-energy B2g mode becomes apparent
("leaks") in the $aa$ scattering geometry, breaking down the aforementioned
selection rules. Similarly, four Ag modes "leak" in the spectra recorded in
the $ac$ scattering geometry at low temperature. Such leakage strongly
suggests a reduction of the D2h point group symmetry of the orthorhombic phase
and, akin to Ta2NiSe5 [10], the most natural point group candidate would
correspond to a monoclinic C2h phase that breaks the $m_{a}(a\rightarrow-a)$
and $m_{c}(c\rightarrow-c)$ mirror symmetries.

The phonon leakage is an indication of symmetry breaking, and the phonon
softening suggests a tendency towards an orthorhombic-to-monoclinic structural
distortion. A recent XRD experiment finds no clear evidence for a structural
transition [22]. Our in-house XRD investigation could not reveal any signature
of such transition down to 80 K [32]. We note that a refinement of the crystal
structure of Ta2NiS5 at 15 K with data taken at the synchrotron is not yet
fully conclusive and calls for additional studies to clarify this issue [32].

Nonetheless, another noteworthy feature of the Raman spectra of Ta2NiS5 is the
anomalous softening of the two lowest-energy B2g modes (hereafter labelled
B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$) on cooling. As previously noted [20],
this behavior is inconsistent with the conventional anharmonic decay of
phonons in solids and might indicate the proximity of a phase transition.
Because of the lattice contraction upon cooling, the corresponding Grüneisen
parameter $\gamma_{i}=-(\Delta\omega_{i}/\omega_{i})/(\Delta V/V)$ of these
phonons is negative, which indicates that the orthorhombic structure is
generally unstable against a reduction of the unit cell volume. Interestingly,
the order parameter of the orthorhombic-to-monoclinic transition in Ta2NiSe5
has the B2g symmetry and the vibration pattern of the two anomalous modes
consists in a shear displacement of the Ta along the a-direction, reminiscent
of the structural distortion (in contrast with the B${}_{2g}^{(3)}$ mode which
involves mostly S vibrations and behaves normally). Taken together, these
observations indicate that lattice strain tuning might be a promising tuning
knob of the electronic properties of this system, as discussed below.

Figure 1:  Log-scale Raman spectra in the $aa$ (a) and $ac$ (b) scattering
geometry, respectively. The spectral features of phonon-mode "leakage" are
labelled by dashed lines.

###  III.2 Strain-dependent Raman spectra

Next, we investigate the strain response of the system. Because of the
anisotropic structure of Ta2NiS5, and the fact that the putative structural
distortion of the material is of B2g symmetry which transforms as an $ac$
quadrupole, we choose to focus on the spectral response of tensile strain
along the crystallographic $a$ direction, which is also favoured by the
morphology of the single crystals.

In Fig. 2, we show the phonon spectra measured at 15 K as function of $a$-axis
tensile strain up to 0.25%. As previously explained, in the monoclinic phase,
all the modes observed in $aa$ [Fig. 2 (a)] or $ac$ [Fig. 2 (b)]
configurations have the Ag symmetry of C2h group. However, we observe a clear
qualitative difference of the strain response for the modes which have larger
intensity in the $aa$ configuration (which had the Ag symmetry of D2h group in
the high-temperature orthorhombic phase) and those seen in the $ac$ scattering
geometry that had B2g character at high temperature. For clarity, we continue
to label these phonons according to their high temperature symmetry.
Strikingly, the B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$ harden significantly and
their lineshape narrows with increasing tensile strain [Insets of Fig. 2 (b)].
This behavior starkly contrast with the behavior of the B${}_{2g}^{(3)}$ mode
or with the observation in the $aa$ configuration, where we can barely detect
a change in the frequency or linewidth of the Ag phonons in the investigated
strain range.

Figure 2: Low-energy Raman spectra measured under various $a$ axis tensile
strain at 15 K in the (a) aa and (b) ac scattering geometries. The inset in
(a) exhibits the Razorbill strain cell CS220T used for the Raman experiments.
The insets in (b) show the strain dependence of the two lowest-frequency
B2g-symmetry phonon modes.

In order to quantify the magnitude of these effects, we have fitted the
different modes using standard Lorentzian lineshape. The strain dependence of
the frequency and linewidth at 15 K for the three B2g phonon modes are shown
in Fig. 3 (a-c). With 0.25 % strain, the relative frequency shift for the
B${}_{2g}^{(1)}$, B${}_{2g}^{(2)}$, and B${}_{2g}^{(3)}$ modes are 5.5, 3.4,
and 0.1 %, respectively. The relative frequency changes for the Ag modes are
all less than 0.5 %. Besides the significant hardening of frequency, the
B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$ modes exhibit a 20% reduction of
linewidth. This effect is completely absent for the other modes, whose
linewidth is nearly strain-independent. The behaviors of the linewidth
demonstrate that within the probed volume the strain is homogeneous, because
inhomogeneity would result in lineshape broadening [39].

Figure 3: Tensile strain ($\epsilon_{aa}$) dependence of the frequency and
full width at half maximum (FWHM) for the three B2g phonon modes at 15 K (a-c)
and 200 K (d-f). The dashed lines are a guide to the eyes.

The relative frequency change, 5.5 % and 3.4 %, of the two lowest-energy B2g
phonon modes at 0.25% strain is anomalously large compared to other systems.
To fix ideas regarding the amplitude of this effect, we provide hereafter a
few examples from the recent litterature for comparison. We first consider the
silicon T2g-symmetry phonon mode which has a frequency of around 520 cm-1.
Under as large as 8.5 % strain induced in silicon membranes by ion
implantation, the magnitude of the relative frequency change is 4.9 % [40],
which is in between the values 5.5 % and 3.4 % we found. We then consider some
two-dimensional systems whose strain response is actively studied [41]. The
phonon modes of monolayer transition-metal-dichalcogenides WSe2 and WS2 only
show around 2% frequency shift at 2.85% tensile strain [39], and CrSBr flake
exhibits about 2% frequency change at 1.7% tensile strain [42]. These ratios
between the frequency change and strain are closer to the strain effect
observed for the $A_{g}$ phonon modes of Ta2NiS5, and generally expected from
Grüneisen’s law, with Grüneisen parameters of individual phonons lying in the
range of a few units for most systems semiconductor and metallic systems [43].
Here and in agreement with the temperature dependent experiment discussed in
the previous section, the Grüneisen parameters for the B${}_{2g}^{(1)}$ and
B${}_{2g}^{(2)}$ phonon modes in Ta2NiS5 are not only much larger in magnitude
but also anomalously negative, reaching -22 and -13.6, respectively.

To find out whether the anomalous behavior of the B${}_{2g}^{(1)}$ and
B${}_{2g}^{(2)}$ modes could be related to strain-induced changes in the
electronic structure, we also studied the strain-dependence of the electronic
excitations. As previously reported [19], the presence of a direct
semiconducting gap suppresses every low-energy electronic excitation in
Ta2NiS5. The lowest interband transition manifests itself already at 200 K in
the form of a broad excitation centered around 300 meV [Fig. 4 (b)]. At 15 K,
the spectrum sharpens significantly and the overall profile changes [Fig. 4
(a)]. We do not see the B2g-symmetry exciton mode as clearly sharp and strong
as in Ref. [19], presumably because of the narrow resonance profile of this
feature (our measurement is performed with 633 nm laser excitation rather than
with 647 nm line used in Ref. [19]). The spectra are however perfectly
consistent with previous reports, showing as well a weaker spectral feature at
around 330 meV, which has previously been interpreted as the second state of
the Rydberg series [19]. Under applied $a$-axis stress, a continuous hardening
of the electronic excitation spectra arises both at high and low temperatures.
With 0.25% strain, a relative blue shift of the excitation spectra amounting
to 4.3% (from 302 to 315 meV) is observed. This 4.3% change of band gap at
0.25% strain is now well comparable to the behaviors encountered in other
semiconducting systems such as bilayer MoS2 under tensile strain [44], or
trilayer MoS2 under various biaxial compressive strains [45]. Qualitatively,
we can associate this behavior with a reduction, under tensile strain along
the crystallographic $a$ direction, of the overlap between Ta 5d orbitals (so
their hybridization), on the one hand, and between the Ni 3d orbitals on the
other hand, which yields an overall reduction of the bandwidth of the
respective bands across the direct semiconducting gap, naturally yielding its
increase. At the same time, the concomitant anomalous hardening of the two
B${}_{2g}^{(1)}$ and B${}_{2g}^{(2)}$ phonon frequencies indicates an
increased stability of the structure at larger unit cell volumes, hinting
towards an intrinsic structural instability of the shrunk unit cell (e.g.
under the effect of cooling).

Figure 4: High-energy Raman spectra measured under various $a$ axis tensile
strain at (a) 15 K and (b) 200 K in the $ac$ scattering geometry.

###  III.3 First-principles calculations and discussion

To gain further insights, it is instructive to compare our experimental
results with those from first principle calculations. In Table 1, we compare
the experimentally measured energy of three optical phonon modes and the band
gap with the calculated values (note that experimental lattice parameters are
used for the calculation [32]). Interestingly, within our computational
approach, the eigenvalue of the dynamical matrix for the B${}_{2g}^{(1)}$ mode
is imaginary, and the structure therefore unstable against the
eigendisplacement of this mode, which precisely involve the shear motion of
the Ta ions along the a axis that would yield a monoclinic distortion akin to
the Se compound. Leaving this aside, we can look at the electronic structure.
As discussed in the supplemental material [32] and in agreement with previous
computational work [23], correlation corrections for the Ni $d$ states need to
be introduced in the framework of the GGA + $U$ approach to open a direct
semiconducting gap in the electronic structure. A 126 meV gap is obtained for
$U$ = 10 eV, which is significantly lower than the experimental value
indicating that non-local self energy effects, especially that of Ta and Ni
orbitals near the Fermi level, should be taken into account to reproduce the
realistic band gap [23]. Nevertheless, the relative magnitude of the strain-
induced effects on the gap is correctly captured by our calculation,
validating the qualitative picture drawn earlier.

We also find that the calculated energy of the phonon modes is in good
agreement with the experimental values, and the absolute renormalization of
the B${}_{2g}^{(2)}$ frequency with strain as well as the absence of
renormalization for the A${}_{1g}^{(1)}$ and B${}_{2g}^{(3)}$ (and so
essentially their respective Grüneisen parameters) are perfectly captured by
the calculation. As already emphasized, the B${}_{2g}^{(2)}$ (and
B${}_{2g}^{(1)}$) mode involves a shear motion of Ta ions along a, a pattern
that was previously found theoretically to strongly modulate the
semiconducting gap size [23]. This raises the question to know whether the
anomalous behavior of this mode should then be attributed to the strain-
induced change of the semiconducting gap, or whether it rather arises only
from lattice stretching.

Table 1: Comparison between the experimentally measured (Exp.) energy values and the calculated (Cal.) ones for selected phonon modes and band gap at 15 K. The A${}_{1g}^{(1)}$ mode refers to the A1g phonon with the lowest energy. The experimentally measured band gap is defined as the energy at which the interband excitations have the maximal intensity; the calculated value corresponds to a direct gap at $\Gamma$ point. The values in the brackets give the change compared to the corresponding zero-strain value. The unit for energies is meV. Strain | 0 % | 0.1 % | 0.25 %
---|---|---|---
| Exp. | Cal. | Exp. | Cal. | Exp. | Cal.
Gap | 302 | 126 | 309 | 132 | 315 | 146
|  |  | (+7) | (+7) | (+13) | (+20)
A${}_{1g}^{(1)}$ | 4.99 | 4.79 | 5.01 | 4.78 | 5.01 | 4.81
|  |  | (+.02) | (-.01) | (+.00) | (+.02)
B${}_{2g}^{(2)}$ | 16.49 | 15.18 | 16.66 | 15.41 | 17.05 | 15.73
|  |  | (+.17) | (+.23) | (+.56) | (+.55)
B${}_{2g}^{(3)}$ | 32.86 | 32.46 | 32.87 | 32.50 | 32.83 | 32.49
|  |  | (+.01) | (+.04) | (-.03) | (+.03)

To address this, we can evaluate the relative contributions to the energy
shifts from the change of lattice structure and gap by comparing the results
obtained with different strains but identical gaps (by adjusting $U$). In
Table 2, we show that with 1.0 % strain and $U$ = 7 eV, the gap size is
comparable to the value obtained with 0.1 % strain and $U$ = 10 eV (Table 1),
but 36 % smaller than the value obtained with 1.0 % strain and $U$ = 10 eV. In
agreement with previous discussion, we find that changes in $U$, strain and
gap size affect only marginally the B${}_{2g}^{(3)}$ and A1g-symmetry phonon
mode frequencies. On the other hand, the 36 % gap size reduction yields a 4%
softening of the B${}_{2g}^{(2)}$ mode, whereas the decrease of strain from 1
% to 0.1 % with constant gap value yields a comparatively larger softening of
10% frequency of the B${}_{2g}^{(2)}$ phonon frequency.

We conclude from this that the anomalous behavior of the B${}_{2g}^{(2)}$
phonon, both as function of temperature and of strain, is rooted in an
incipient instability of the orthorhombic lattice structure against a shear
monoclinic distortion of the Ta chains, which is likely to occur irrespective
of the electronic gap size. Additional work is required to demonstrate whether
such this distortion is effectively realized in Ta2NiS5, as suggested by the
Raman data, but our work demonstrates that the susceptibility towards it is
very large, and would likely occur upon compression of the lattice. At the
same time, such perturbation would allow a reduction of the semiconducting gap
size, which would offer an interesting way to approach the regime in which an
EI can occur (i.e. when the gap size becomes lower than the exciton binding
energy).

Table 2: Comparison between the calculated energy values for the B2g phonon modes and band gap under 1.0 % strain. | Gap | B${}_{2g}^{(2)}$ | B${}_{2g}^{(3)}$
---|---|---|---
$U$ = 10 eV | 213.5 meV | 17.21 meV | 32.40 meV
$U$ = 7 eV | 136.7 meV | 16.50 meV | 32.48 meV
Relative change | -36.0 % | -4.1 % | +0.2 %

##  IV Conclusion

In summary, we have used polarization-resolved Raman spectroscopy to study the
response of phononic and electronic degrees of freedom to uniaxial strain in
Ta2NiS5. In particular, we found that tensile strain along the $a$ axis
results in an anomalously large frequency increase for two B2g phonon modes.
The corresponding Grüneisen parameter for the lowest-energy B2g mode is
negative and has an magnitude larger than 20. DFT calculations support that
such anomalous Grüneisen parameter results from a proximity to a structural
instability similar to the one in excitonic insulator candidate Ta2NiSe5.
Besides,the band gap of this system increases with strain, as the tensile
strain reduces the orbital overlap and in turn the bandwidth. These
experimental results establish Ta2NiS5 as a suitable platform to explore
strain tuning of lattice dynamics and electronic structure.

###### Acknowledgements.

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - TRR 288 - 422213477 (project B03), and Projektnummer
449386310. M. F. acknowledges funding from the Alexander von Humboldt
fundation, and the YIG preparation program of the Karlsruhe Institute of
Technology. R.H. acknowledges support by the state of Baden-Württemberg
through bwHPC.

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