††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Vibrational properties differ between halide and chalcogenide perovskite
semiconductors, and it matters for optoelectronic performance

Kevin Ye  Department of Materials Science and Engineering, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  Matan Menahem  Department
of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot
76100, Israel  Tommaso Salzillo  Department of Industrial Chemistry,
University of Bologna, Bologna, Italy  Boyang Zhao  Mork Family Department of
Chemical Engineering and Materials Science, University of Southern California,
Los Angeles, CA 90089, USA  Shanyuan Niu  Mork Family Department of Chemical
Engineering and Materials Science, University of Southern California, Los
Angeles, CA 90089, USA  Florian Knoop  Department of Physics, Chemistry and
Biology (IFM), Linköping University, SE-581 83, Linköping, Sweden.  Ida
Sadeghi  Department of Materials Science and Engineering, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  Olle Hellman  Department of
Physics, Chemistry and Biology (IFM), Linköping University, SE-581 83,
Linköping, Sweden.  Department of Molecular Chemistry and Material Science,
Weizmann Institute of Science, Rehovot 76100, Israel  Jayakanth Ravichandran
Mork Family Department of Chemical Engineering and Materials Science,
University of Southern California, Los Angeles, CA 90089, USA  Ming Hsieh
Department of Electrical and Computer Engineering, University of Southern
California, Los Angeles, CA 90089, USA  Core Center of Excellence in
NanoImaging (CNI), University of Southern California, Los Angeles, CA 90089,
USA  Rafael Jaramillo<EMAIL_ADDRESS>Department of Materials Science and
Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Omer Yaffe<EMAIL_ADDRESS>Department of Chemical and Biological
Physics, Weizmann Institute of Science, Rehovot 76100, Israel

(February 29, 2024)

###### Abstract

We report a comparative study of temperature-dependent photoluminescence and
structural dynamics of two perovskite semiconductors, the chalcogenide BaZrS3
and the halide CsPbBr3. These materials have similar crystal structures and
direct band gaps, but we find that they have quite distinct optoelectronic and
vibrational properties. Both materials exhibit thermally-activated non-
radiative recombination, but the non-radiative recombination rate in BaZrS3 is
between two and four orders-of-magnitude faster than in CsPbBr3. Raman
spectroscopy reveals that the effects of phonon anharmonicity are far more
pronounced in CsPbBr3 than in BaZrS3. Further, although both materials feature
a large dielectric response due to low-energy polar optical phonons, the
phonons in CsPbBr3 are substantially lower in energy than in BaZrS3. Our
results suggest that electron-phonon coupling in BaZrS3 is more effective at
non-radiative recombination than in CsPbBr3, and that BaZrS3 may also have a
substantially higher concentration of non-radiative recombination centers than
CsPbBr3. The low defect concentration in CsPbBr3 may be related to the ease of
lattice reconfiguration, typified by anharmonic bonding. It remains to be seen
to what extent these differences are inherent to the chalcogenide and halide
perovskites, and to what extent they can be affected by materials processing;
comparing BaZrS3 single-crystals and thin films provides reason for optimism.

Photoluminescence, Raman spectroscopy, perovskites, chalcogenides, halides

##  I Introduction

Halide perovskites exhibit outstanding optoelectronic properties and are under
intensive development for photovoltaic (PV) applications [1, 2, 3]. They can
be synthesized at near-room temperature and feature sharp band-edge optical
absorption and long excited-state charge carrier lifetimes [4, 5, 6, 7, 8].
They also suffer from the drawbacks of thermal and chemical instability and
lead toxicity. For these reasons, research continues into alternative
materials for thin-film photovoltaics. Chalcogenide perovskites are an
intriguing alternative. They feature similar crystal structure and range of
direct band gap as their halide counterparts [9, 10, 11]. They also share a
very large dielectric constant, which suggests similar lattice dynamics and
screening of charged-defects [32, 13]. However, chalcogenide perovskites are,
in other ways, quite different. The materials are made of Earth-abundant
elements with little toxicity concern. Synthesis is challenging and typically
requires very high temperature and air-free processing. Once formed, they are
very stable [9, 14, 4, 16, 17]. The optoelectronic properties of chalcogenide
perovskites have been little reported. Existing published results suggest that
photoluminescence (PL) is not strong and that defect control is a challenge,
but long PL transient decay times have been measured [26, 9, 16, 17, 19, 20,
21]. No chalcogenide perovskite thin-film solar cells have been reported,
likely due to the challenging conditions required for film synthesis.

Many researchers, including ourselves, have argued that the outstanding
optoelectronic properties of halide perovskites are intertwined with their
strongly anharmonic structural dynamics [22, 23, 24, 25, 26, 27, 28, 29, 30,
31]. This poses questions of how the optoelectronic properties and lattice
dynamics of chalcogenide and halide perovskites compare and what we can learn
about photovoltaic performance from this comparison.

Here, we employ temperature-dependent PL and Raman scattering spectroscopy,
alongside first-principles self-consistent phonons simulations, to compare two
exemplary single-crystal samples: the chalcogenide perovskite BaZrS3 and the
halide perovskite CsPbBr3. We also briefly discuss results for BaZrS3
epitaxial thin films. BaZrS3 and CsPbBr3 share the same distorted perovskite,
corner-sharing, orthorhombic crystal structure (space group $Pnma$) within the
studied temperature range [32, 33]. Between 80 and 300 K, the PL emission
intensity of both materials decreases due to thermally-activated non-radiative
recombination, with electron-phonon interactions broadening the PL linewidth
in both cases. However, CsPbBr3 emission intensity surpasses that of single-
crystal BaZrS3 by over four orders of magnitude, suggesting a significantly
higher concentration of very-low-barrier non-radiative recombination centers
in BaZrS3. PL emission intensity increases by approximately two orders of
magnitude in the BaZrS3 film, compared to the crystal, but remains well below
that of CsPbBr3.

We use Raman spectroscopy and first-principles simulations to show that the
phonon spectrum in BaZrS3 extends to nearly double the frequency in CsPbBr3.
Raman analysis demonstrates that CsPbBr3 exhibits pronounced anharmonic
characteristics, even at low temperatures. In contrast, BaZrS3 exhibits
predominantly harmonic vibrational behavior akin to traditional tetrahedrally-
bonded semiconductors. Our findings are consistent with the hypotheses that
the lower-frequency and strongly anharmonic structural dynamics in CsPbBr3 may
reduce carrier capture cross sections due to Franck-Condon effects [34]. They
may also decrease defect concentration through annealing, even at room
temperature [35]. Definitive tests of these hypotheses await detailed
calculations of rate coefficients supported by first-principles quantum
calculations and defect spectroscopy to identify and quantify recombination-
active defects experimentally.

##  II  Methods

We grew single crystals of BaZrS3 with dimensions of approximately 100 $\mu$m
using a flux method as reported previously [9]. We grew epitaxial thin films
of BaZrS3 approximately 40 nm thick using molecular beam epitaxy (MBE) as
reported previously [4]. We grew single crystals of CsPbBr3 with dimensions on
the order of 2 mm using the vapor saturation of an antisolvent (VSA) method as
reported previously [36].

We performed Raman scattering in a home-built system in back-scattering
geometry. We used a 1.58 eV CW pump-diode laser (Toptica Inc., USA), which is
below the band gap of BaZrS3 (1.9 eV) [9, 37] and CsPbBr3 (2.32 eV) [38]. We
performed PL spectroscopy using the same optical system and a 2.54 eV CW
sapphire-pumped diode laser (Coherent Inc., USA). The signal was measured in a
1 m long spectrometer (FHR1000, Horiba Inc.) equipped with a holographic
grating (1800 gr/mm for Raman and 150 gr/mm for PL) and a Synapse Plus CCD
detector (Horiba Inc.).

The sample temperature was controlled using a liquid-N2\- or liquid He-cooled
high-vacuum optical cryostat (Janis Inc., USA) for $T\leq 400$ K. For higher
temperature measurements, the temperature was controlled in an optical furnace
(Linkam TS1000) under N2 flow. We performed measurements for a series of
increasing temperatures. We took precautions to isolate the effect of sample
temperature on PL intensity by maintaining stable measurement conditions,
refocusing the laser before every measurement, and taking repeated
measurements across the whole temperature range. The measurement system and
protocols are further described in Refs. [39, 40, 41, 42, 22].

We calculate normalized PL yield ($Y$) by analyzing the integrated PL
intensity, including both the band-to-band peaks and the shoulders. The PL
data is measured in counts, and we compute intensity by dividing by the
acquisition time. We define $Y$ as the ratio between the PL intensity emitted
from the sample ($I_{PL}$, measured in counts/sec) and the excitation laser
intensity that reaches the interior of the sample ($I_{in}$):

$Y=\frac{I_{PL}}{I_{in}}=\frac{I_{PL}}{I_{0}}(1-R_{s})~{}.$ (1)

We measure the laser intensity $I_{0}$ (in photons/sec) at the microscope
entrance. We compute $I_{in}$ from $I_{0}$ by accounting for losses due to
reflection from the cryostat windows and the sample surface. The same cryostat
window reflection losses affect the collected PL intensity and cancel in the
expression for $Y$, leaving only a dependence on the sample surface power
reflection coefficient ($R_{s}$). Further details are presented in Sec. S6 in
the Supplemental Material (SM) [44].

##  III Results

###  III.1 Photoluminescence and Non-Radiative Recombination

In Fig. 1, we present temperature-dependent PL spectra of BaZrS3 and CsPbBr3
measured between 80–300 K. At low temperatures, spectra are dominated by band-
edge recombination at 1.96 eV and 2.34 eV for BaZrS3 and CsPbBr3,
respectively. The low-temperature spectra for both samples feature low-energy
shoulders, which are quenched at elevated temperatures. The band-to-band peak
and the shoulder are much broader in BaZrS3 than in CsPbBr3. In both
materials, PL is quenched with increasing temperature, and for BaZrS3, it
becomes too weak to measure above 170 K. To eliminate external features in the
PL spectra, the high-temperature spectra were subtracted from those in Fig. 1.

Figure 1: Temperature-dependent PL spectra. PL spectra measured between 80 K
and 300 K for (a) BaZrS3 and (b) CsPbBr3, after subtraction of the background
spectra at 170 K and 300 K, respectively. The line color indicates
temperature. BaZrS3 and CsPbBr3 were measured with different pump laser
fluence: 4.3 MJ/cm2 for BaZrS3, and 2.7 $\times$10-4 MJ/cm2 for CsPbBr3.
Figure 2: Model-based analysis of normalized PL yield ($Y$, Eq. (2)) and
linewidth ($\Gamma$). (a) $Y$ as a function of the inverse temperature. (b)
Fit to a model with a second non-radiative recombination pathway.
$E_{b}/k_{B}$ fall below our measurement temperature window. (c) $\Gamma$ of
the band-to-band peak as a function of the temperature. All fitted values can
be found in Sec. S6 of the SM [44]. Purple and orange traces are for CsPbBr3
and BaZrS3, respectively.

BaZrS3 and CsPbBr3 are direct band-gap semiconductors and strongly absorb the
pump laser. At the pump laser energy of 2.54 eV, the absorption coefficient of
BaZrS3 is 1.68$\times\text{{10}}^{5}$ $\text{{cm}}^{-1}$, and that of CsPbBr3
is 4.04$\times\text{{10}}^{5}$ $\text{{cm}}^{-1}$ [4, 3]. Therefore, the pump
laser absorption depth is much smaller than the thickness for both single-
crystal samples, and all the light entering the samples is absorbed. However,
the PL emission intensity of BaZrS3 dwarfs that of CsPbBr3.

In Fig. 2(a), we present an Arrhenius plot of the normalized PL yield ($Y$).
We fit the data to a minimal model of thermally-activated non-radiative
recombination [46, 47]:

$Y(T)=\frac{1}{1+Ae^{-E_{a}/k_{B}T}+Be^{-E_{b}/k_{B}T}}~{},$ (2)

$E_{a}$ and $E_{b}$ are the activation energies of two non-radiative
recombination mechanisms, and $A=k_{nr,a}^{0}/k_{r}$ and
$B=k_{nr,b}^{0}/k_{r}$ are ratios of exponential prefactors that compare non-
radiative to radiative recombination rates. As discussed below, two non-
radiative recombination mechanisms are required for a minimal model of the
data. This model assumes that all radiative recombination has a sufficiently
small activation energy that appears barrierless within the measurement
window. The effect of this assumption is that the PL internal quantum
efficiency (IQE) approaches unity at low temperatures. The best-fit model
parameters are presented in Table 1.

Table 1: Best-fit parameters for PL yield, Eq. (2). Results are presented here for $E_{b}/k_{B}=8$ K. In Table S3 in the SM [44], we present results for other choices of $E_{b}/k_{B}$. Material |  $I_{0}$ (photons/sec) |  $E_{a}$ (meV) |  $E_{b}/k_{B}$ (K) | A | B
---|---|---|---|---|---
BaZrS3 | 6.7 $\times\text{{10}}^{15}$ | 68 $\pm$ 7 | 8 | (7.2 $\pm$ 4.0) $\times\text{{10}}^{17}$ | (3.1 $\pm$ 0.5) $\times\text{{10}}^{14}$
CsPbBr3 | 4.3 $\times\text{{10}}^{11}$ | 86 $\pm$ 5 | 8 | (2.9 $\pm$ 0.8) $\times\text{{10}}^{13}$ | (1.1 $\pm$ 0.2) $\times\text{{10}}^{10}$

We find that BaZrS3 and CsPbBr3 feature a non-radiative recombination process
with similar activation energy $E_{a}=68\pm 7$ and $86\pm 5$ meV,
respectively. This is apparent by eye in the Arrhenius plot (Fig. 2(a)), as
both data sets have similar curvature and a knee at approximately the same
temperature. However, the PL yield of CsPbBr3 is over four orders of magnitude
larger than that of BaZrS3, and the data do not extrapolate towards the same
yield in the low-temperature limit. The presence of non-radiative
recombination mechanisms with similar activation energy belies the difference
in PL intensity.

To explain the divergence in PL intensity, we invoke a second non-radiative
recombination mechanism in BaZrS3, with activation energy $E_{b}$ too low to
be determined in our measurement window. We also assume a similar second
mechanism in CsPbBr3 for consistency in the model function. By construction,
the thermal scale $E_{b}/k_{B}$ falls well below our experimental temperature
window, and therefore, $E_{b}$ cannot be determined by regression. This low-
temperature process is saturated within the measured temperature range, and
$e^{-E_{b}/k_{B}T}\approx 1$. With this assumption, we fix the temperature
scale $E_{b}/k_{B}$ at a low value and fit the data to determine $B$. In Fig.
2(a, inset), we present a family of model functions with $E_{b}$ varied
manually; for $E_{b}$ sufficiently small, its value has an insignificant
impact on the fit results and our conclusions. The ratio
$B_{\text{{BZS}}}/B_{\text{{CPB}}}$ is meaningful, as it determines the
measured difference in low-temperature PL yield between BaZrS3 (BZS) and
CsPbBr3 (CPB). We find that:

$\frac{B_{\text{BZS}}}{B_{\text{CPB}}}=\frac{\left(k_{nr,b}^{0}/k_{r}\right)_{\text{BZS}}}{\left(k_{nr,b}^{0}/k_{r}\right)_{\text{CPB}}}=28,181~{}.$
(3)

Both materials have comparable optical absorption coefficients near their band
edge; therefore, we can approximate
$\left(k_{r,\text{CPB}}/k_{r,\text{BZS}}\right)\approx\mathcal{O}(1)$. We
therefore conclude that the rate $k_{nr,b}^{0}$ of low- (or zero-) barrier
non-radiative recombination is $\mathcal{O}(10^{4})$ times faster in BaZrS3
than in CsPbBr3.

We analyze the PL linewidth ($\Gamma$) for insight into how excited states
couple to lattice vibrations. In Fig. 2(b), we show the temperature dependence
of $\Gamma$ of the main PL peak at $E_{g}$ for each material. We use multiple
Gaussian peaks to fit the data, three for CsPbBr3 and four for BaZrS3. For
both materials, the main PL peak is well modeled by two Gaussian peaks, and we
determine $\Gamma$ numerically from the best-fit models. We fit $\Gamma$ to a
relation describing thermal and non-thermal broadening, previously applied to
halide perovskites [38]:

$\Gamma_{PL}(T)=\Gamma_{0}+\frac{\gamma}{e^{E_{ph}/k_{B}T}-1}~{},$ (4)

$\Gamma_{0}$ represents temperature-independent contributions to $\Gamma$,
such as heterogeneous broadening due to quenched structural disorder. The
second term describes broadening due to coupling between excited states and
lattice vibrations with characteristic energy $E_{ph}$ and electron-phonon
coupling constant $\gamma$. These lattice vibrations are likely longitudinal
optical phonons (see below), and the electron-phonon interactions can be
described through deformation potentials.

We find that luminescent band-edge states couple to vibrations with
characteristic energy that is similar in both materials: $E_{ph}=$ 24 $\pm$ 7
meV for BaZrS3, and 17 $\pm$ 4 meV for CsPbBr3 (see below). The electron-
phonon coupling constant $\gamma$ is larger for BaZrS3 (200 $\pm$ 100 meV)
than for CsPbBr3 (50 $\pm$ 10 meV). $\gamma$ is related to deformation
potentials and polar-optical coupling. The higher value in BaZrS3 may be
related to the stronger covalent nature of the Zr-S bonds compared to Pb-Br
bonds: 32% covalent for Zr-S, compared to 26% for Pb-Br, according to the
Pauling scale. This implies enhanced sensitivity of electron energies to bond
angles in the chalcogenide, as we have studied previously [48, 13]. We also
see that BaZrS3 has larger heterogeneous broadening, $\Gamma_{0}$, than
CsPbBr3, reflecting a higher concentration of structural defects and other
forms of non-thermal disorder. In addition to affecting the PL spectra, these
factors affect how steeply optical absorption rises at the band gap, the width
of the Urbach tail, and the phonon-limited charge transport mobility, and are
therefore relevant to photovoltaic performance [49, 50, 51].

###  III.2 Raman Scattering and Phonon Dynamics

The lower luminescence yield and larger PL broadening in BaZrS3, compared to
CsPbBr3, are related to differences in lattice vibrations. To better
understand these relationships, we turn to theoretical and experimental
studies of the phonon spectra. In Fig. 3, we present the total (black) and
partial (orange, green, and blue) vibrational density of states of BaZrS3
(top) and CsPbBr3 (bottom), calculated using the temperature-dependent
effective potential (TDEP) method at 300 K (details in Sec. S7 the SM [44, 8,
7, 5, 17, 6, 9, 16, 20, 10, 19, 11, 15, 12, 14, 13, 18]). The optical
vibrations in BaZrS3 span frequencies up to 430 $\text{cm}^{-1}$ (53 meV),
while those in CsPbBr3 peak at 160 $\text{cm}^{-1}$ (20 meV). This is due to
the higher mass of the atoms in CsPbBr3 and the more covalent character of the
bonds in BaZrS3 [68].

LO phonons likely contribute most to the temperature-dependent PL linewidth
broadening. In Fig. 3, the vertical red lines provide a measure of the phonon
LO character: As detailed in Sec. S7 in the SM [44], the LO character is
estimated by computing the longitudinal charges for optical vibrations from
the mode eigenvectors and atomic Born charge tensors projected on the [101]
excitation direction and divided by the vibration frequency to account for the
average mode amplitudes. We see that BaZrS3 has many phonons with prominent LO
character along the [101] excitation direction, whereas CsPbBr3 only shows one
mode of appreciable LO character at $\approx 125\,{\rm cm}^{-1}$. This factor
likely contributes to the stronger temperature-dependent PL broadening in
BaZrS3 in addition to the higher deformation potentials.

The characteristic energy of lattice vibrations responsible for PL broadening
in CsPbBr3, $E_{ph}\simeq\text{{17}}$ meV ($\approx 140\>\text{cm}^{-1}$), is
comparable to the most prominent LO phonon in the calculated spectrum. For
BaZrS3, $E_{ph}\simeq\text{{24}}$ meV ($\approx 190\>\text{cm}^{-1}$) falls
roughly in the middle of the calculated spectrum. Likely, band-edge states
couple to a diversity of lattice vibrations in both materials through polar-
optical coupling and deformation potentials, and the characteristic energy
determined by PL broadening represents a distribution of processes. Similar
conclusions have been reached in studies of ultra-fast lattice rearrangement
following optical excitation of halide perovskites, including CsPbBr3 [69].

Figure 3: Calculated room-temperature phonon density of states (DOS, right
axis) of BaZrS3 and CsPbBr3. The colors indicate the partial contribution of
the A (blue), M (green), and X (orange) ions in the AMX3 formula. The red
lines are the measure (left axis) of LO character. Figure 4: Temperature-
dependent Raman scattering spectra of BaZrS3 and CsPbBr3. Temperature is
indicated by color. The individual data sets are shifted vertically for
clarity; the dashed horizontal lines indicate the zero level for each data
set.

In Fig. 4, we present the low-frequency Raman spectra of BaZrS3 and CsPbBr3
measured in the temperature range 80–350 K. The spectra of BaZrS3 and CsPbBr3
are consistent with previous reports [42, 24]. In Sec. S8 in the SM [44], we
present the Raman spectra of BaZrS3 measured over a wider temperature range,
from 10–875 K, showing no indication of a phase transition [25, 27, 28].

The Raman spectra of CsPbBr3 exhibit sharp first-order peaks (one photon
scattered by one vibration) [23] at low temperatures, which shift and broaden
significantly with increasing temperature. These structural dynamics are
characteristic of halide and certain oxide perovskites [21, 76, 77, 78]. The
broad, featureless spectra at high temperatures, combined with a rapid
decrease in vibrational lifetime (evidenced by peak broadening), indicate the
impact of anharmonic vibrational dynamics [23, 38].

At low temperatures, the Raman spectra of BaZrS3 resemble those of CsPbBr3 and
other halide perovskites [39, 79], with a similar number of prominent peaks
divided into doublets and triplets. The vibrations in BaZrS3 are higher in
frequency than those in CsPbBr3. Unlike CsPbBr3, the Raman peaks in BaZrS3
remain sharp and distinct with increasing temperature, and even at 350 K,
there is scant signature of anharmonic dynamics. The harmonic dynamics in
BaZrS3 recall those in tetrahedrally-bonded semiconductors (e.g. GaAs), in
which vibrations along the stiff covalent bonds do not interact and change
little with temperature [80, 81].

Figure 5: Analyzis of Raman Spectra: (a) Frequencies ($\Omega$ \- top) and
linewidths ($\Gamma$ \- bottom) of the first-order peaks of BaZrS3 (left) and
CsPbBr3 (right) as a function of temperature. All values were normalized to
the value at 80 K. Different markers mark different peaks. (b) Linear
regression results (R2 \- top, and slopes $\alpha$ \- bottom) of the
temperature-dependent frequencies (left) and linewidths (right) of BaZrS3 and
CsPbBr3, between 80-280 K. The results are plotted as a function of peak
frequency at 80 K, the lowest temperature we measured, where the phonon
population and anharmonicity are the lowest. Orange and purple markers are for
BaZrS3 and CsPbBr3, respectively.

We can more quantitatively compare the structural dynamics of BaZrS3 and
CsPbBr3 by considering the temperature dependence of the phonon frequencies
and linewidths. We model the Raman spectra as a convolution of pseudo-Voigt
peaks multiplied by the Bose-Einstein occupation factor [29, 30, 31] (see Sec.
S8 in the SM [44]). We focus on the prominent peaks because they yield the
most confident fits. In Fig. 5(a) we present the vibrational frequency
($\Omega$) and linewidth ($\Gamma$) of the prominent first-order peaks of
BaZrS3 (left panel, orange) and CsPbBr3 (right panel, purple). The frequency
and linewidth data are normalized to their values at 80 K for ease of
comparison (for original values, see Sec. S8 in the SM [44]). The Raman peaks
of BaZrS3 red shift and broaden linearly with increasing temperature, as is
expected from perturbation theory for weak anharmonic effects [85, 86]. A
slight change in slope near 200 K may relate to the observed increase in the
low-frequency dielectric susceptibility in this same temperature range [32].

CsPbBr3 exhibits a similar monotonic and linear trend in frequency and
linewidth with increasing temperature up to 280 K, above which the data
deviate substantially, including sharp blue shifts and rapidly increasing
linewidth. We attribute these changes to two causes: (1) the thermal
population of all the optical phonons above $T_{\text{{Debye}}}\approx 250$ K,
and (2) the incipient orthorhombic-to-tetragonal phase transition at $T\approx
360$ K. Peak broadening with increasing temperature is attributed to the
shortening of vibrational lifetime due to phonon-phonon scattering [85, 86,
30]. It is apparent that the structural dynamics of BaZrS3 exhibit minor
thermal effects and are less impacted by anharmonic vibrational interactions
than the dynamics of CsPbBr3 [85, 86].

The extent of anharmonicity in the phonon dynamics can be further quantified
by analyzing the deviation from linearity of the temperature dependence of the
phonon frequencies and linewidths. A linear temperature dependence is expected
within perturbation theory for weak phonon-phonon interactions. Therefore,
strong and anharmonic interactions are indicated by deviations from linearity.
In Fig. 5(b), we present the R2 parameter and the absolute value of the slope
($\lvert\alpha\rvert$) of linear regression to the $\Omega(T)$ and $\Gamma(T)$
fit results (for mean values, see Sec. S8 in the SM [44]). For CsPbBr3, the R2
values are far from unity, and the slopes are much larger than for BaZrS3.
This analysis further supports the conclusion that anharmonic effects in
BaZrS3 can be modeled as a weak perturbation, whereas in CsPbBr3, they
substantially change the lattice vibrational properties.

##  IV Discussion

Lead halide perovskites excel as PV absorbers largely due to their slow rates
of defect-assisted Shockley-Read-Hall (SRH) non-radiative free energy loss.
Electron-phonon interactions mediate SRH recombination processes, converting
excited-state energy into phonons and heat. Therefore, by focusing on
vibrational properties and electron-phonon interactions, we can better
understand the origin of the excellent PV performance of the halides and
identify promising new semiconductors for thin-film PV.

Chalcogenide perovskites have an untested potential for PV largely due to
challenges in thin film synthesis. There are few published measurements of the
optoelectronic properties of chalcogenide perovskites, and the results are
ambiguous. We reported time-resolved PL data that suggest long radiative
lifetimes in BaZrS3 and the Ruddlesden-Popper compound Ba3Zr2S7, comparable to
lead halides and other high-performing PV materials such as CIGS [19]. Several
research groups have reported band-edge PL measured from BaZrS3 samples of
various form factors, including solution-processed nanoparticles and thin
films made by sulfurizing oxide precursors [21, 20, 26, 16]. However,
quantitative PL measurements (albeit on early-stage powder samples) suggest
low quantum efficiency and low quasi-Fermi level splitting (i.e. open-circuit
voltage potential) [9]. Our experience of measuring PL on BaZrS3 single
crystals and thin films is that most samples have weak or no measurable band-
edge PL and that the band-edge PL can be strongly position-dependent. Strong
sample-to-sample variation, combined with challenges in synthesis and defect
control, make it difficult to assess the technology potential of chalcogenide
perovskites. The comparison here between halide and chalcogenide model
systems, focusing on fundamentals of lattice vibrations and electron-phonon
interactions, provides some clarity and suggests a way forward for materials
development.

SRH recombination is much faster in BaZrS3 than in CsPbBr3 due to an
unidentified recombination pathway that is active even at low temperature,
implying that it has a very low thermal activation energy (Fig. 2(b)). The
theory of multi-phonon non-radiative recombination highlights the importance
of the overlap between the vibrational wavefunctions of the initial and final
electronic states, consistent with the Franck-Condon principle [34]. Model
calculations for a harmonic lattice indicate a steep dependence of SRH
recombination rate on phonon energy [87]. Lower phonon energy means more
phonons are needed to participate in non-radiative recombination processes.
Within this model, the overlap of vibrational wavefunctions is suppressed with
increasing phonon numbers. Therefore, the lower phonon frequency in CsPbBr3
compared to BaZrS3 may explain the difference in SRH recombination rates, even
if the concentration and properties of defects are similar.

The same harmonic model also predicts that SRH recombination rates increase
with the extent of lattice distortion accompanying defect charge capture; this
distortion can be parameterized by the Hwang-Rhys factor [87, 34]. These
lattice distortions often result in long-lived charge trapping and are
responsible for persistent photoconductivity in many semiconductors; it is
related to DX-center phenomena in III-V compounds, and we termed it defect-
level switching in the context of resistive switches [88, 89, 90, 91]. There
have been many studies of persistent photoconductivity in halide perovskites.
Still, as of yet, there is no clear model for defects in CsPbBr3 [92, 93, 94,
95]. The timescale of photoconductivity decay in BaZrS3 crystals depends on
sample polishing, suggesting that defects introduced during plastic
deformation (e.g. dislocations) may be effective charge traps with large
Hwang-Rhys factors [96]. However, these observations may not be directly
relevant to the Hwang-Rhys factors of recombination-active defects, which may
differ from the traps that influence photoconductivity decay. Further, it has
been suggested that polaronic effects resulting from strong charge-lattice
coupling - including Fröhlich (i.e. large, effective mass states) polarons and
defect-localized polarons - may suppress SRH recombination rates by screening
dissimilar charges [97, 98]. It is likely that a full quantum calculation of
electron-phonon interactions, combined with first-principles modeling of the
known crystal structures and the most likely recombination-active defects,
will be required to resolve these questions over the role of strong electron-
phonon coupling on SRH recombination rates.

The effect of anharmonic phonon-phonon interactions on SRH recombination
remains an open question. It would be enlightening to compare a model
calculation of vibrational wavefunction overlap during multi-phonon
recombination with and without anharmonic lattice vibrations and for varying
Hwang-Rhys factor for otherwise similar phonon energies. Model system
calculations could help to understand the conflated effects of low phonon
energy, lattice distortion at defects, and anharmonic vibrations on SRH
recombination rates. However, full quantum calculations are likely needed to
accurately model the differences in carrier capture rates between halide and
chalcogenide perovskites.

The discussion to this point has focused on the effects of lattice vibrations
on SRH recombination rates via the carrier capture processes. However, another
significant unknown is the concentration of recombination-active defects.
Anharmonic vibrations indicate softer bonds, a more pliable crystal lattice,
and lower activation energy for ionic diffusion [99, 100, 101]. Therefore,
anharmonic vibrations in lead halide perovskites may improve optoelectronic
performance by accelerating the elimination of point defects by annealing,
even at room temperature (termed “self-healing”) [35, 102, 103, 104, 105].
BaZrS3, in contrast, requires high temperature to form and, therefore, is
likely to retain quenched-in defects (e.g. chemical impurities). This suggests
that the PL yield and PV performance of chalcogenide perovskites will depend
more strongly on defect control during sample processing than the performance
of halide perovskites.

Defect characterization by methods such as deep-level transient spectroscopy
(DLTS) and related forms of impedance spectroscopy could directly measure
defect concentrations and quantify the extent of self-healing [106, 107, 108,
109, 110, 111, 112]. Interpretation of DLTS data is challenging for halide
perovskites due to mixed ionic-electronic conduction, but progress has been
reported [113]. Similar studies have not yet been undertaken for chalcogenide
perovskites.

In the meantime, we are encouraged by empirical evidence that the fast SRH
recombination rates evidenced here for BaZrS3 crystals can be reduced through
sample processing. In Figs. S3-S4 in the SM [44], we present the results of PL
measurements on a BaZrS3 epitaxial thin film. The film PL intensity is two
orders of magnitude higher than that of the crystal, and the results of
fitting the data using Eq. (2) suggest that the concentration of
recombination-active defects is reduced by as much as 10,000 times in the film
compared to the crystal (see Sec. S6.d in the SM [44] for a detailed
discussion). This result demonstrates that the low PL yield of the BaZrS3
crystal presented here is not intrinsic to the material and that the PL yield
and optoelectronic performance can be improved through changes in materials
synthesis and processing. We look forward to systematic investigations of
performance-limiting defects, in concert with continuous advances in materials
synthesis, in a rational effort to improve materials performance - as has been
historically required for other semiconductors.

##  V Conclusions

We have presented a detailed spectroscopic comparison between two perovskite
semiconductors, the chalcogenide BaZrS3 and the well-studied halide CsPbBr3.
We find that the PL emission yield of CsPbBr3 is over four orders of magnitude
higher than that of BaZrS3 when comparing single crystal samples, indicative
of a far higher concentration of very-low-barrier, non-radiative recombination
centers in BaZrS3. MD simulations show that BaZrS3 exhibits higher LO
vibrational frequencies than CsPbBr3, which couple to charge carriers and
increase carrier capture cross sections due to Franck-Condon effects [34].
Raman scattering results suggest that the strongly anharmonic structural
dynamics in CsPbBr3, relative to BaZrS3, may reduce defect concentration
through accelerated solid-state diffusion and annealing [35]. The usefulness
of BaZrS3 (and likely other chalcogenide perovskites) for optoelectronics will
hinge on continued progress in defect control during synthesis and processing.

## Acknowledgements

We acknowledge the support of Dr. I. Pinkas (WIS) for help designing the
experimental setup, Dr. L. Segev (WIS) for developing the Raman software, and
Dr. S. Aharon (Princeton) for help in the synthesis of CsPbBr3 crystals. OY
acknowledges funding from the European Research Council starting grant (850041
- ANHARMONIC). We acknowledge support from the MIT-Israel Zuckerman STEM Fund
and the Sagol Weizmann-MIT Bridge Program. We acknowledge support from the
United States-Israel Binational Science Foundation, grant no. 2020270. We
acknowledge support from the National Science Foundation (NSF) under grant no.
1751736. We acknowledge support from the Skolkovo Institute of Science and
Technology and the MIT-Skoltech Next Generation Program. KY acknowledges
support from the NSF Graduate Research Fellowship, grant no. 1745302. B.Z. and
J.R. acknowledge support from an ARO MURI with award number W911NF-21-1-0327,
an NSF grant with award number DMR-2122071, and an ONR grant with award number
N00014-23-1-2818. F.K. acknowledges support from the Swedish Research Council
(VR) program 2020-04630, and the Swedish e-Science Research Centre (SeRC). The
computations were enabled by resources provided by the National Academic
Infrastructure for Supercomputing in Sweden (NAISS) at NSC and PDC partially
funded by the Swedish Research Council through grant agreement no. 2022-06725.

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Supplemental Material
Vibrational properties differ between halide and chalcogenide perovskite
semiconductors, and it matters for optoelectronic performance

## S6 Photoluminescence Data and Analysis

### S6.a PL Data Processing

Figure S1 shows the as-measured photoluminescence (PL) spectra of BaZrS3
(left) and CsPbBr3 (right) in the temperature range of 80–300 K. The
measurement parameters are listed in Table S2.

Figure S1: Photoluminescence Raw Spectra of BaZrS3 (left) and CsPbBr3 (right) as measured, from 80–300 K. Table S2: Photoluminescence Measurement Parameters Parameter | BaZrS3 | CsPbBr3
---|---|---
Excitation Wavelength [nm] | 488 | 488
Excitation Energy [eV] | 2.54 | 2.54
Excitation Power [mW] | 2.74 | 1.74$\times\text{{10}}^{-4}$
Acquisition Time [s] | 15 | 5
Spot Size [$\mu$m] | 1.1 | 1.1
Excitation Fluence [MJ/cm2] | 4.3 | 2.7 $\times$10-4

We first subtract a wavelength-independent constant background from each
spectrum from electronic noise to ensure that the tails of the spectra
approach zero. In the case of CsPbBr3, this was the only pre-processing step
performed before integrating the entire spectra. In the case of BaZrS3, due to
its low PL intensity, we found a small but non-negligible contribution from
the excitation laser, reaching a maximum near 2.2 eV. The PL signal from the
sample is quenched at $\approx$170 K, leaving only the temperature-independent
laser contribution. Therefore, we remove it by subtracting the spectrum
measured at high temperature from all lower-temperature data.

We model the band-to-band PL intensity by fitting four Gaussian peaks to the
spectra for BaZrS3 and three in the case of CsPbBr3. Figure S2 shows an
example of a fit at 80 K.

Figure S2: Photoluminescence intensity modeling. Gaussian deconvolution
results for the photoluminescence spectra of BaZrS3 (left) and CsPbBr3 (right)
at 80 K. The blue and green traces are the data and fit, respectively. The
shaded traces are individual peaks – purple and orange for the main transition
and other transitions, respectively.

### S6.b Determination of Normalized PL Yield

To convert the measured spectra in counts per second to normalized PL yield
($Y$, Eq. (1) [1]), we start by considering the incident excitation laser
power measured upstream of the microscope ($I_{0}$). The cryostat windows
attenuate the incident laser light and the sample PL emission, and these
effects cancel in the expression for $Y$. We calculate sample reflection loss
using the Fresnel equation; with $\sqrt{\epsilon}=n$, we obtain $n=3.1$ for
BaZrS3 and $n=2.3$ for CsPbBr3 [2, 3].

We fit the normalized PL yield data to Eq. (2) for fixed values of $E_{b}$, as
described in the main text [1]. In Table S3, we report the excitation power
$I_{0}$ and the best-fit values.

Table S3: Normalized PL Yield Fitted Parameters |  $I_{0}$ (photons/sec) |  $E_{a}$ (meV) |  $E_{b}/k_{B}$ (K) | A | B
---|---|---|---|---|---
BaZrS3 |  6.7$\times\text{{10}}^{15}$ |  68 $\pm$ 7 | 0 |  (6.9 $\pm$ 3.8) $\times\text{{10}}^{17}$ |  (2.8 $\pm$ 0.5) $\times\text{{10}}^{14}$
|  6.7$\times\text{{10}}^{15}$ |  68 $\pm$ 7 | 2 |  (6.9 $\pm$ 3.8) $\times\text{{10}}^{17}$ |  (2.9 $\pm$ 0.5) $\times\text{{10}}^{14}$
|  6.7$\times\text{{10}}^{15}$ |  68 $\pm$ 7 | 4 |  (7.0 $\pm$ 3.9) $\times\text{{10}}^{17}$ |  (3.0 $\pm$ 0.5) $\times\text{{10}}^{14}$
|  6.7$\times\text{{10}}^{15}$ |  68 $\pm$ 7 | 8 |  (7.2 $\pm$ 4.0) $\times\text{{10}}^{17}$ |  (3.1 $\pm$ 0.5) $\times\text{{10}}^{14}$
CsPbBr3 |  4.3$\times\text{{10}}^{11}$ |  86 $\pm$ 5 | 0 |  (2.8 $\pm$ 0.8) $\times\text{{10}}^{13}$ |  (9.6 $\pm$ 1.4) $\times\text{{10}}^{9}$
|  4.3$\times\text{{10}}^{11}$ |  86 $\pm$ 5 | 2 |  (2.8 $\pm$ 0.8) $\times\text{{10}}^{13}$ |  (9.9 $\pm$ 1.5) $\times\text{{10}}^{9}$
|  4.3$\times\text{{10}}^{11}$ |  86 $\pm$ 5 | 4 |  (2.9 $\pm$ 0.8) $\times\text{{10}}^{13}$ |  (1.0 $\pm$ 0.2)$\times\text{{10}}^{10}$
|  4.3$\times\text{{10}}^{11}$ |  86 $\pm$ 5 | 8 |  (2.9 $\pm$ 0.8) $\times\text{{10}}^{13}$ |  (1.1 $\pm$ 0.2) $\times\text{{10}}^{10}$

### S6.c Determination of PL Linewidth

We require two considerably overlapping Gaussian peaks to model the band-to-
band PL lineshape of both materials. Therefore, we numerically calculate the
linewidth (full-width-half-maximum, $\Gamma$) of these combined peaks. We fit
the results using Eq. (4) [1], and the best-fit parameters are presented in
Table S4.

Table S4: PL Linewidth Fitted Parameters |  $E_{ph}$ (meV) |  $\gamma$ (meV) |  $\gamma_{0}$ (meV)
---|---|---|---
BaZrS3 Crystal | 24 $\pm$ 7 | 205 $\pm$ 116 | 57 $\pm$ 4
BaZrS3 Film | 10 $\pm$ 4 | 66 $\pm$ 28 | 121 $\pm$ 7
CsPbBr3 Crystal | 18 $\pm$ 4 | 48 $\pm$ 12 | 9 $\pm$ 2

### S6.d Results for Epitaxial BaZrS3 Film

In addition to the single crystals represented in the main text [1], we
measured the PL of a BaZrS3 thin film grown by molecular beam epitaxy [4]. We
present the data and analysis in Fig. S3 and S4. In Table S5, we compare
between the best-fit values to Eq. (2) [1]. The PL data and thermal trends are
markedly different in the film and the crystal. Both exhibit band-edge PL at
nearly the same energy and the PL intensity is suppressed with increasing
temperature. However, the film PL intensity is stronger than that of the
crystal. Also, the film PL persists even at 400 K, whereas it is quenched in
the crystal above 170 K. The persistence of PL intensity to higher
temperatures in the film allows us to observe a red shift of the band gap with
increasing temperature. This trend becomes notable above 300 K.

This difference in PL yield is reflected in the modeling results. Like the
crystals, the BaZrS3 film exhibits a thermally-activated non-radiative
recombination process with a characteristic energy of $46$ meV. Also, as for
the crystal, we need to invoke a low-barrier, non-radiative recombination
process to model the film PL yield data using Eq. (2) [1]. However, the value
of $B$ is approximately one to four orders of magnitude smaller compared to
the BaZrS3 crystal.

The simple interpretation of the fit results is that the defects responsible
for the two non-radiative recombination processes are $\approx$101–104 times
less concentrated in the film. Likely, the film and the crystal have different
types of recombination-active defects, and the two-mechanism model is an
oversimplification. However, even without detailed knowledge of the types of
defects present, we can conclude that changes in BaZrS3 sample processing can
produce large differences in PL yield and, by extension, optoelectronic
performance.

Notably, the film PL is more intense than that of the crystal because the film
is thin and attenuates the excitation laser intensity by only $40\%$. This
value assumes two bounces of light through a 40 nm thick film, and the
absorption coefficient at 488 nm (1.68 $\times\text{{10}}^{5}$ cm-1) is
determined from previous work [4].

Figure S3: PL Spectra of Epitaxial BaZrS3 Film. PL spectra of BaZrS3 film (right) and crystal (left). The color scale indicates temperature. The laser excitation power ($I_{0}$) is indicated as well. Figure S4: Analysis of Epitaxial BaZrS3 Film Photoluminescence Normalized photoluminescence yield of BaZrS3 Film compared to the BaZrS3 crystal and CsPbBr3 crystal shown in the main text. Table S5: Normalized PL Yield Best-Fit Parameters for Film Sample |  $I_{0}$ (photons/sec) |  $E_{a}$ (meV) |  ${E_{b}}/{k_{b}}$ (K) | A | B
---|---|---|---|---|---
BaZrS3 Crystal | 6.7$\times\text{{10}}^{15}$ | 68 $\pm$ 7 | 0 | (6.9 $\pm$ 3.8) $\times\text{{10}}^{17}$ | (2.8 $\pm$ 0.5) $\times\text{{10}}^{14}$
BaZrS3 Film | 4.6$\times\text{{10}}^{14}$ | 46 $\pm$ 7 | 0 | (2.0 $\pm$ 0.4) $\times\text{{10}}^{13}$ | (6.9 $\pm$ 0.2) $\times\text{{10}}^{12}$
CsPbBr3 Crystal | 4.3$\times\text{{10}}^{11}$ | 86 $\pm$ 5 | 0 | (2.8 $\pm$ 0.8) $\times\text{{10}}^{13}$ | (9.6 $\pm$ 1.4) $\times\text{{10}}^{9}$

## S7 Phonon density of states and LO character

### S7.a Self-consistent phonon scheme

We use the _stochastic temperature-dependent effective potential_ (sTDEP)
scheme to compute temperature-dependent phonons described by the force
constants $\\{\Phi_{ij}(T)\\}$ which depend on temperature $T$. This scheme is
formally equivalent to the QSCAILD method described in Ref. [5] and relies on
minimizing the mean-square error of the harmonic model forces,

$\displaystyle S[\Phi,T]$
$\displaystyle\equiv\sum_{i}\left\langle(f_{i}-f_{\Phi;i})^{2}\right\rangle_{\Phi,T}~{},$
(S1)

where $\langle\cdot\rangle_{\Phi,T}$ denotes a thermal average with respect to
the force constants $\Phi$ at temperature $T$, $i,j$ denote atomic and
Cartesian indices, and the harmonic force $f_{\Phi;i}$ is given by
$f_{\Phi;i}=-\sum_{j}\Phi_{ij}u_{j}$, where $u_{j}$ is a vector with atomic
displacements from the thermal average positions.

As shown and discussed in Ref. [5, 6], the least-squares solution to Eq. (S1)
is given by

$\displaystyle\Phi_{ij}(T)=-\left\langle
u_{i}u_{j}\right\rangle_{\Phi,T}^{-1}\left\langle
f_{i}u_{j}\right\rangle_{\Phi,T}^{\phantom{-1}}\equiv\left\langle\frac{\partial^{2}V}{\partial
u_{i}\partial u_{j}}\right\rangle_{\Phi,T}~{},$ (S2)

i.e., the configuration average of the second derivative of the potential
energy $V$ with respect to the effective harmonic distribution of atomic
positions at target temperature $T$. This solution is equivalent to a Gibbs-
Bogoliubov free energy minimization and yields the best trial free energy of
harmonic form. Further details are given in [7, 8, 9].

After obtaining the temperature-dependent force constants at room temperature,
we can use them to compute the dynamical matrix

$\displaystyle D_{ij}({\bf q})=\frac{1}{\sqrt{m_{i}m_{j}}}\sum_{L}{\rm
e}^{{\rm i}{\bf q}\cdot{\bf R}_{L}}\Phi_{i0,jL}~{},$ (S3)

via mass weighting with the atomic masses $m_{i}$ and Fourier-transforming to
q-space by summing over lattice vectors ${\bf R}_{L}$, from which temperature-
renormalized dynamical properties can be inferred. In particular, we define
the eigenvalue equation diagonalizing the atomic basis $i,j$ to mode indices
$s$,

$\displaystyle\sum_{j}D_{ij}({\bf q})\epsilon_{j{\bf q}s}=\omega^{2}_{{\bf
q}s}\epsilon_{i{\bf q}s}^{\phantom{I}}~{},$ (S4)

with eigenfrequencies $\omega_{{\bf q}s}$ and eigenvectors $\epsilon_{i{\bf
q}s}$.

Without going into details, we note that these frequencies do _not_ correspond
to the quasiparticle poles of phonon excitations. For physical spectral
properties, phonon-phonon interactions need to be taken into account, which
can be done in the TDEP framework by obtaining third-order force constants
[10]. We denote the fully thermally shifted quasiparticle frequencies as
$\tilde{\omega}_{{\bf q}s}$ in the following.

### S7.b Density of states

From the phonon quasiparticle spectrum $\\{\tilde{\omega}_{{\bf q}s}\\}$ at
room temperature, we can compute the _density of states_ (DOS)

$\displaystyle{\rm DOS}(\omega)=\sum_{{\bf
q}s}\delta(\omega-\tilde{\omega}_{{\bf q}s})~{}.$ (S5)

### S7.c LO character

Longitudinal optical (LO) modes are characterized by a longitudinal charge,

$\displaystyle Z^{\rm LO}_{{\bf k}s}=\frac{{\bf k}}{k}\cdot{\bf Z}_{{\bf
k}s}~{},$ (S6)

with

$\displaystyle{\bf Z}_{{\bf k}s}=\sum_{i}Z_{i}\epsilon_{i{\bf
k}s}/\sqrt{m_{i}}~{},$ (S7)

where $Z_{i}$ is the Born effective charge tensor for atom $i$, and ${\bf
k}\to 0$ denotes a small but finite wave vector characteristic for the probing
laser in an optical experiment.

The longitudinal charge creates a macroscopic polarization along $\bf k$ when
mode $s$ is excited

$\displaystyle{\bf p}^{\rm LO}_{{\bf k}s}=Z^{\rm LO}_{{\bf k}s}A_{{\bf
k}s}^{\phantom{I}}\frac{{\bf k}}{k}~{},$ (S8)

where $A_{{\bf k}s}$ is the mode amplitude. Through this polarization, a mode
$s$ can couple to electronic excitations via Fröhlich coupling. Since the mode
amplitude goes with

$\displaystyle\langle A_{{\bf k}s}^{2}\rangle\propto 1/\omega_{{\bf
k}s}^{2}~{},$ (S9)

we take

$\displaystyle Z^{\rm LO}_{{\bf k}s}/\omega_{{\bf k}s}$ (S10)

as an estimate for the “longitudinal mode coupling strength” or simply “LO
character”. Since this coupling strength is direction-dependent via the
incident wave vector $\bf k$, we take $\bf k$ to point in $[101]$ direction in
line with the experimental setup. However, we have verified that the
qualitative picture does not change if $\bf k$ is chosen along another
direction, e.g., one of the Cartesian axes.

### S7.d Computational details

For both systems, we use a unit cell with 20 atoms, space group 60 (_Pnma_)
(Materials Project codes mp-540771 for BaZrS3 and mp-567681 for CsPbBr3 [11]).
We use the temperature-dependent effective potential method (TDEP) with the
stochastic sampling scheme described above (sTDEP). Details can be found in
[8] and references therein. For force sampling, 3x2x3 supercells (360 atoms)
and the all-electron DFT code FHI-aims were used, with _light default_ basis
sets [12], and am05 _xc_ functional [13, 14]. To compute the Born effective
charge tensor, Quantum Espresso v7.1 was used [15, 16, 17], with 75 Ry energy
cutoff, PBE functional [18], and Schlipf-Gygi norm-conserving pseudopotentials
created with the ONCVPSP code [19, 20].

## S8 Raman Data and Analysis

### S8.a BaZrS3 Raman Data Over a Wide Temperature Range

In Fig. S5, we present the Raman spectra of BaZrS3 from 10–875 K, where we see
signs of sample degradation in the optical microscope. Each spectrum was
normalized between zero and one by reducing a constant background and dividing
by the maximum intensity.

Figure S5: Raman Spectra of BaZrS3 crystal in the range of 10–875 K. Spectra
were normalized and shifted vertically for clarity.

The Raman spectra below 80 K were measured in the same optical system, using
liquid-He as the cryogenic coolant.

At low temperatures, the Raman spectra are characterized by sharp isolated
peaks assigned to first-order scattering from single normal-mode vibrations
(one photon scattering by one vibration). As often happens, the first-order
peaks slightly red shift and broaden with increasing temperature [21, 22]. The
broad, higher-frequency ($\omega\geq 300$ cm-1) features that emerge with
increasing temperature are assigned to second-order scattering due to their
absence at low temperatures, their rapid increase with temperature [23], and
from the calculated phonon density of states. A broad background forms below
the low-frequency peaks; this we assign to disorder-enhanced high-order
scattering of unknown origin [21]. We can distinguish first-order scattering
peaks from the broad features at higher temperatures only by following the
peaks at increasing temperatures from cryogenic conditions. Further assignment
using integrated intensity temperature dependence is presented below.

The temperature-dependent Raman data show no evidence of a phase
transformation; BaZrS3 remains in the orthorhombic phase throughout [24]. This
is consistent with reported calorimetric measurements [25] and x-ray studies
[26, 27, 28] showing no phase transformation up to $\approx$ 1300 K, at which
point degradation begins.

### S8.b Deconvolution

To assign and analyze the temperature-dependent trends of the Raman peaks,
each spectrum was deconvolved into a product of the Bose-Einstein distribution
($n_{\omega,T}$) and multiple pseudo-Voigt oscillators:

$\begin{split}\mathcal{S}(\omega)=&\left[n_{\omega,T}+1\right]\times\\\
&\sum_{j}c_{j}\left[(1-\zeta_{j})\frac{\omega|\Omega_{j}|\Gamma_{j}^{2}}{\omega^{2}\Gamma_{j}^{2}+(\omega^{2}-\Omega_{j}^{2})^{2}}+\zeta_{j}\exp{\left(\frac{-(\omega-\Omega_{j})^{2}\ln{16}}{\Gamma_{j}^{2}}\right)}\right]~{},\end{split}$
(S11)

$\mathcal{S}(\omega)$ and $\omega$ are the Raman intensity and Raman shift,
and $\Omega_{j}$, $\Gamma_{j}$ and $c_{j}$ are the $j$’th mode’s vibrational
frequency, FWHM, and intensity, respectively. The Bose-Einstein distribution
is given by $n_{\omega,T}=(e^{\hbar\omega/k_{B}T}-1)^{-1}$, where $\hbar$,
$k_{B}$ and $T$ are the reduced Planck’s constant, the Boltzmann constant and
temperature. The parameter $0\leq\zeta_{j}\leq 1$ is the Gaussian fraction of
the pseudo-Voigt line shape, used to capture broadening unrelated to
vibrational lifetime, such as higher-order scattering. Overall,
$\zeta_{j}\approx 0$ was used for the sharp first-order peaks.

Figure S6 shows the deconvolution results for the 80 K spectra of BaZrS3
(left) and CsPbBr3 (right). The blue and green traces are the measured and
fitted spectra, while the red traces are individual peaks. The intensity of a
Raman peak depends on temperature through the Bose-Einstein distribution
according to the scattering order [23]. As the temperature is lowered, first-
order peaks are more prominent than higher orders. Therefore, we identify the
first-order scattering peaks by their presence in the spectra at 10 K (black
traces), which contain primarily first-order peaks. The inset shows the
deconvolution of the high-frequency modes of BaZrS3, which we assign as
second-order scattering. This assignment is corroborated by the calculated
vibrational density of states, which shows a high density of optical modes up
to 250 cm-1 and a few optical modes at 300 cm-1 and 400 cm-1, with a large gap
between them. Assuming low Raman activity of the 300 cm-1 and 400 cm-1
branches, our assignment nicely matches the calculated vibrational density of
states.

In the case of CsPbBr3, the sharp peaks at 80 K broaden significantly at
higher temperatures, making it possible to determine their scattering order
only at low temperatures (see Fig. 4 [1]).

Figure S6: Raman deconvolution. pseudo-Voigt deconvolution results for the
Raman spectra of BaZrS3 (left) and CsPbBr3 (right) at 80 K. The blue and green
traces are the data and fit, respectively. The red traces are individual
peaks. Black traces are the spectra at 10 K that were used to identify the
first-order peaks at higher temperatures. Inset - deconvolution of the high-
frequency modes of BaZrS3.

### S8.c Assignment of Scattering Order

To substantiate our assignment of the modes of BaZrS3, we compare the
temperature dependence of the integrated intensity to the expected dependence
of first- and second-order scattering [21]. The intensity of first-order Raman
scattering is proportional to the product of the Bose-Einstein distribution
and the mode spectral function ($\mathcal{J}_{j}$) [29]. Therefore, for Stokes
scattering, it follows that [30, 31]

$\mathcal{S}_{j}(\omega)=\Xi\int_{0}^{\infty}e^{-i\omega t}\left\langle
U^{*}_{j},U_{j}\right\rangle~{}\text{{d}}t=\Xi~{}[n_{\omega,T}+1]~{}\mathcal{J}_{j}(\Omega_{j},\Gamma_{j})~{},$
(S12)

where $\Xi$ is the proportionality factor that contains the Raman tensor and
properties of the material affecting the scattering, and $\mathcal{J}_{j}$ is
defined as the temperature-independent Fourier transform of the pair
displacement correlation function ($\left\langle
U^{*}_{j},U_{j}\right\rangle$) such that

$\int_{0}^{\infty}\mathcal{J}_{j}~{}\text{{d}}\omega=1~{}.$ (S13)

Therefore, we expect the integrated intensity of any pseudo-Voigt peak
originating from first-order scattering to remain constant at all
temperatures. The intensity of second-order Raman scattering (one photon
scattered by two vibrations of opposite momentum) is more complicated to
handle because, at each frequency, there are contributions from many
combinations of modes. Therefore, we can not describe the expected dependence
of a specific mode and must calculate the dependence of an extensive spectral
range. To do that, we model second-order scattering as the convolution of mode
spectra functions [21],

$\begin{split}\mathcal{S}(\omega)&=~{}\Xi\int_{0}^{\infty}e^{-i\omega
t}\sum_{j,j^{\prime}}\left\langle
U^{*}_{j},U^{*}_{j^{\prime}},U_{j},U_{j^{\prime}}\right\rangle~{}\text{{d}}t\\\
&=~{}\Xi~{}\times\sum_{j,j^{\prime}}[n_{\omega,T}+1]\mathcal{J}_{j}(\Omega_{j},\Gamma_{j})*[n_{\omega,T}+1]\mathcal{J}_{j^{\prime}}(\Omega_{j^{\prime}},\Gamma_{j^{\prime}})~{},\end{split}$
(S14)

following the same notation and conditions from Eq. (S12) and Eq. (S13). The
set of $\mathcal{J}_{j}$ that reproduces the experimental line shape is not
unique; However, if we find one solution that can reproduce the line shape and
integrated intensity, we prove that second-order scattering dominates in the
spectral range in question. To that end, we estimate that combination modes
($j\neq j^{\prime}$) have negligible contribution compared to overtones
($j=j^{\prime}$), and simplify Eq. (S14) to a single sum,

$\mathcal{S}(\omega)=\Xi~{}\times\sum_{j}[n_{\omega,T}+1]\mathcal{J}_{j}(\Omega_{j},\Gamma_{j})*[n_{\omega,T}+1]\mathcal{J}_{j}(\Omega_{j},\Gamma_{j})~{}.$
(S15)

Figure S7: pseuo-Voigt integrated intensities compared to expected trends. (a)
low frequency ($\omega<300$ cm-1) sharp peak compared to expectation for
first-order scattering. (b) - high-frequency ($\omega\geq 300$ cm-1) peaks
compared to expectation for second-order scattering. (c) - low-frequency broad
features ($\Gamma_{j}>10$ cm-1) compared to expectation for first- and second-
order scattering. Colored markers represent the different fitted peaks with a
dashed line to guide the eye, and the dotted gray lines are expectation
curves. All curves were normalized to their values at 80 K for clarity.

The simplified version allows us to solve each $\mathcal{J}_{j}$ separately
and then sum the contributions. As mentioned, we assign all the high-frequency
peaks ($\omega\geq 300$ cm-1) to second-order Raman scattering. Therefore, we
tried to find a set of $\mathcal{J}_{j}$ to reproduce the line shape of the
high-frequency spectral range and compare the temperature dependence of the
integrated intensities, as follows:

  1. 1.

Draw the high-frequency pseudo-Voigt peak using the fitted parameters at
$T_{0}$.

  2. 2.

Find a $\mathcal{J}_{j}$ such that
$\mathcal{S}_{pV}(\omega)=\cfrac{[n_{\omega,T_{0}}+1]\mathcal{J}_{j}(\Omega_{j},\Gamma_{j})*[n_{\omega,T_{0}}+1]\mathcal{J}_{j}(\Omega_{j},\Gamma_{j})}{[n_{\omega,T_{0}}+1]}$
and $\int_{0}^{\infty}\mathcal{J}_{j}~{}\text{{d}}\omega=1$.

  3. 3.

Integrate the fitted pseudo-Voigt peaks and the sum of reproduced line shapes
separately.

  4. 4.

Assuming the set of $\mathcal{J}_{j}$ is temperature-independent, recalculate
the expected line shape at $T\neq T_{0}$ for the given set through the Bose-
Einstein distribution.

  5. 5.

Compare the temperature-dependent integrated intensity of the fitted peaks and
the recalculated line shapes.

Figure S7 shows the integrated intensities of the low-frequency sharp peaks
(a) assigned as first-order scattering, the high-frequency peaks (b) assigned
as second-order scattering, and the low-frequency broad background peaks
($\Gamma_{j}>10$ cm-1) compared to the expected trends (gray lines) for first-
and second-order scattering (c).

Figure S8: Analysis of Raman Spectra: (a) $\Omega_{j}$ (top) and $\Gamma_{j}$
(bottom) of the first-order peaks of CsPbBr3 (left) and BaZrS3 (right) as a
function of temperature. (b) Linear regression results (R2 \- top, and slopes,
$alpha$ \- bottom) of $\Omega_{j}(T)$ (left) and $\Gamma_{j}(T)$ (right) of
CsPbBr3 and BaZrS3, between 80–280 K. A different color represents each peak.

The integrated intensity of the low-frequency sharp modes varies by $\approx
50\%$ within the temperature range, which is within our allowed error range,
considering the complexity of a real crystal and the experiment. Furthermore,
according to our analysis of second-order Raman scattering, there is an
excellent correlation between the integrated intensity of the high-frequency
peaks and the expectation. The expected curve was drawn using fitting
parameters from the 80 K fit. Nevertheless, using the fitting parameters of
other temperatures produced similar results (data not shown). The case of the
broad low-frequency features presented in the right panel is more complicated
- their integrated intensity remains constant for $T\leq 200~{}K$, then
increases dramatically with temperature, almost following a second-order
scattering trend. The dramatic change correlates well with a reported increase
in dielectric response at kHz frequencies [32]. Therefore, we faithfully
assign the sharp $\omega<300$ cm-1 modes and the $\omega\geq 300$ cm-1 peaks
to first- and second-order scattering, respectively. Following previous work
[21], we assign these features to high-order scattering due to symmetry-
breaking.

### S8.d Linear Trends

Figure S8(a) shows the $\Omega_{j}$ (top) and $\Gamma_{j}$ (bottom) of the
prominent first-order peaks of CsPbBr3 (left) and BaZrS3 (right) as a function
of temperature, resulting from the pseudo-Voigt fit. Figure S8(b) shows the
results of the linear regression of $\Omega_{j}(T)$ (left) and $\Gamma_{j}(T)$
(right) of CsPbBr3 (circles) and BaZrS3 (diamonds) as a function of the peak
index $j$. The R2 value and slope are presented in the top and bottom panels,
respectively.

Table S6 shows the average R2 values and slopes of the linear regression of
$\Omega_{j}(T)$ and $\Gamma_{j}(T)$ of the peaks of BaZrS3 and CsPbBr3 in the
range of 80–280 K. The average R2 values and their standard deviation (STD)
indicate that the frequencies and linewidths of BaZrS3 are more linear than
those of CsPbBr3, showing average values closer to 1 and lower STD. Moreover,
the average slope values indicate that the peaks of CsPbBr3 shift and broaden
more than the peaks of BaZrS3. The drastic change in the structural dynamics
and the anharmonicity of CsPbBr3 compared to BaZrS3 are indicated by the
larger STDs.

Table S6: Mean values of the linear regression presented in Fig. S8 Material | Property | Parameter | Mean $\pm$ STD
---|---|---|---
BaZrS3 | $\Omega_{j}(T)$ | R2 | $0.98\pm 0.01$
|  | Slope [cm K]-1 | $0.020\pm 0.008$
| $\Gamma_{j}(T)$ | R2 | $0.9\pm 0.1$
|  | Slope [cm K]-1 | $0.015\pm 0.006$
CsPbBr3 | $\Omega_{j}(T)$ | R2 | $0.8\pm 0.3$
|  | Slope [cm K]-1 | $0.03\pm 0.02$
| $\Gamma_{j}(T)$ | R2 | $0.83\pm 0.09$
|  | Slope [cm K]-1 | $0.05\pm 0.03$

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