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

# The Disorder Origin of Raman Scattering In Perovskites Single Crystals

Matan Menahem  Department of Chemical and Biological Physics, Weizmann
Institute of Science, Rehovot 76100, Israel  Nimrod Benshalom  Department of
Chemical and Biological Physics, Weizmann Institute of Science, Rehovot 76100,
Israel  Maor Asher  Department of Chemical and Biological Physics, Weizmann
Institute of Science, Rehovot 76100, Israel  Sigalit Aharon  Department of
Molecular Chemistry and Material Science, Weizmann Institute of Science,
Rehovot 76100, Israel  Roman Korobko  Department of Chemical and Biological
Physics, Weizmann Institute of Science, Rehovot 76100, Israel  Sam Safran
Department of Chemical and Biological Physics, Weizmann Institute of Science,
Rehovot 76100, Israel  Olle Hellman  Department of Molecular Chemistry and
Material Science, Weizmann Institute of Science, Rehovot 76100, Israel  Omer
Yaffe<EMAIL_ADDRESS>Department of Chemical and Biological
Physics, Weizmann Institute of Science, Rehovot 76100, Israel

###### Abstract

The anharmonic lattice dynamics of oxide and halide perovskites play a crucial
role in their mechanical and optical properties. Raman spectroscopy is one of
the key methods used to study these structural dynamics. However, despite
decades of research, existing interpretations cannot explain the temperature
dependence of the observed Raman spectra. We demonstrate the non-monotonic
evolution with temperature of the scattering intensity and present a model for
2nd-order Raman scattering that accounts for this unique trend. By invoking a
low-frequency anharmonic feature, we are able to reproduce the Raman spectral
line-shapes and integrated intensity temperature dependence. Numerical
simulations support our interpretation of this low-frequency mode as a
transition between two minima of a double-well potential surface. The model
can be applied to other dynamically disordered crystal phases, providing a
better understanding of the structural dynamics, leading to favorable
electronic, optical, and mechanical properties in functional materials.

Dynamic disorder, perovskites, Raman scattering, anharmonicity, second order,
double well

## Introduction

Lattice dynamics and structural phase transitions in perovskite crystals have
been extensively studied for decades. [1, 2, 3] The high coordination number
for cations in the perovskite structure (12) offers a range of bonding
strengths that leads to strongly anharmonic ionic thermal motion. [4, 5]
Importantly, such anharmonic motion enables desirable and tunable electro-
mechanical and optoelectronic responses. [6, 7]

Raman scattering spectroscopy is a powerful tool in the study of vibrational
properties of materials, and was extensively used to investigate the
structural dynamics of perovskites. [8, 9, 10, 11, 12] However, the
interpretation of their Raman spectrum and its evolution with temperature has
proven very challenging. The anharmonic character leads to inadequacy of the
Raman selection rules, as they are dictated by the average symmetry of the
crystals. [13, 12, 14, 15] This is most apparent in the cubic phase
($Pm\bar{3}m$) of perovskite crystals, which is 1st-order Raman inactive and
yet, typically displays a large Raman cross-section. [16, 17] Despite the many
similarities in their crystal structure and observed spectra, existing
interpretations for the Raman scattering of oxide and halide perovskites are
markedly different. Moreover, these models fail to capture or explain several
important experimental observations.

In the case of oxide perovskites, it is widely accepted that the Raman
spectrum in the cubic phase is dominated by two-phonon Raman scattering, [18,
16, 17, 19] associated with phonons at high symmetry points in the Brillouin
zone. [20, 21, 22] However, the intensity of the scattered light is unusually
strong and does not scale with temperature as two-phonon scattering dictates.
[23, 24, 25, 26]

The Raman spectra of halide perovskites are dominated by an over-damped peak
centered at zero-frequency (termed ’central peak’). [27, 28] It was suggested
that the central peak is related to polar fluctuations [19, 26, 29] between
short-lived distorted structures, [30, 31, 32, 33] giving rise to 1st-order
Raman scattering. While it explains the observed Raman and is backed up by
molecular dynamics simulations, [29, 28] it contradicts the Raman selection
rules derived from the average symmetry. Moreover, this picture is at odds
with two additional experimental observation. The first is that the halide
perovskites have a very sharp absorption edge (i.e. low Urbach energy) in the
visible range, which implies a highly ordered crystal structure. [34, 35] The
second is that the ’central peak’ is observed in Raman scattering but not in
THz-absorption measurements. [36, 37, 38]

Figure 1: Temperature-dependent Raman spectra of cubic BaTiO3, SrTiO3 and
CsPbBr3. a) Spectra (from left to right) of BaTiO3, SrTiO3 and CsPbBr3. The
spectra of BaTiO3 and SrTiO3 are divided to two scattering regimes, showing
the non-monotonic temperature dependence. Arrows emphasize trends with
increasing temperature. b) Normalized Stokes integrated intensity of BaTiO3
(green), SrTiO3 (red) and CsPbBr3 (yellow). In grey is the expected
temperature dependence from conventional two phonon scattering. The deviation
from the expected temperature dependence is evident.

In this study we propose a unified description for the Raman scattering of
both oxide and halide perovskites that explains its unique temperature
dependence, as well as resolves the problems described above. Our model
attempts to capture the symmetry broken structures using a picture consistent
with the average symmetry and its constraints.

First, we measure the temperature evolution of the Raman spectra of BaTiO3,
SrTiO3, and CsPbBr3 crystals, observing an unusual trend with temperature, in
strike contrast to harmonic Raman theory. Next, motivated by the double-well
(DW) picture of an atomic potential surface in cubic halide perovskites, [39]
we simulate the dynamics of a single particle in 1D potential and find it
leads to excess Raman scattering intensity. To reconcile our experimental
observations with the simulated results, while maintaining the cubic symmetry,
we present a 2nd-order Raman scattering model for dynamically disordered
crystals. The core concept of our model is a localized low-frequency
fluctuation that embodies all anharmonic dynamics, and participates in light
scattering. By allowing the potential surface to change with temperature, our
model is able to explain the Raman spectra of oxide and halide perovskites.

## Results and Discussions

The Raman spectra of BaTiO3, SrTiO3 and CsPbBr3 at selected temperatures in
the cubic phase are shown in Fig. 1(a). We collected the Raman spectra in a
back-scattering configuration, with special care taken to isolate the effect
of temperature on the intensity, as described in the Methods. The data of
BaTiO3 and CsPbBr3 begin after the tetragonal-cubic phase transition (405 K).
The data of SrTiO3 begin at 300 K since the phase transition is at 105 K. [40]
The full data set, including the spectra of SrTiO3 below 300 K, is depicted in
Fig. S1 and S2 in the Supplemental Material. Figure 1(b) shows the Stokes
integrated intensity as function of temperature for BaTiO3 (green), SrTiO3
(red) and CsPbBr3 (yellow) along with the expected trend for 2-phonon
scattering (gray). In particular, we observe a non-monotonic trend with
temperature, with a clear maximum for SrTiO3 and BaTiO3 and subsequent
decrease in scattering intensity. We therefore divide the spectra of for
SrTiO3 and BaTiO3 in Fig. 1(a) into two regimes according to the trend in the
Stokes integrated intensity, below and above the maximum intensity temperature
($T_{m}$).

The effect of temperature on the spectral line-shape and integrated intensity
in the oxide perovskites is different above and below $T_{m}$: SrTiO3 exhibits
peak broadening and increase in integrated intensity; in BaTiO3 a sharp
central peak appears at the phase transition [41] and decreases with
temperature (black arrow in Fig. 1(a)), causing only slight changes to the
integrated intensity (polarized spectra of the BaTiO3 central peak are
depicted in Fig. S3 in the SM). The spectra of CsPbBr3 exhibit a similar trend
to BaTiO3, with no decrease in the central peak intensity, and only $10\%$
increase in integrated intensity. Due to the similar crystal structures and
Raman spectra, we speculate a decreasing integrated intensity would have also
been observed in CsPbBr3, had the crystals remained intact above 700 K. The
decrease in integrated intensity is in stark contrast to the harmonic theory
of Raman scattering. [42] Conventionally, the Stokes integrated intensity is
expected to increase with temperature due the to thermal population of
vibrations.

Contemporary explanations for the unconventional properties of halide
perovskites [43] are based on a DW potential model. Frozen phonon calculations
have shown the existence of a DW potential around instability points in the
Brillouin zone [31, 44, 39] as well as a 2D ’saddle’ potential related to the
phase diagram. [30] To test the effect of a DW potential on the Raman
scattering and whether it can account for the decrease in integrated
intensity, we numerically simulated the Raman spectra of a single particle in
a DW potential.

Figure 2:  Numerical simulation results for a particle in a potential well. a)
Probability density for the location of the particle at various temperatures
in a DW potential. b) Simulated 2nd-order Raman spectra of a particle in a
double-well (blue) and parabolic (red) potential at $T=500~{}\text{{Ha}}$. c)
Integrated intensity of 2nd-order inelastic spectra as function of temperature
for a double-well (blue) and parabolic (red) potential. d) Excess intensity at
$T=500~{}\text{{Ha}}$ as function of the barrier height
($H_{\text{{Barrier}}}=V(0)-V(W)$) showing positive monotonic dependence.
Green shading emphasizes the excess intensity.

### Numerical Simulation

We numerically solve the time-independent Schr$\ddot{\text{{o}}}$dinger
equation for a single particle in a 1D DW potential:

$\left(\frac{-\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}+V\left(x\right)\right)\psi\left(x\right)=E\psi\left(x\right)~{},$ (1)

where $E$ and $\psi(x)$ are the real eigenvalues and eigenstates,
respectively. Because the cubic phase is Raman inactive, the decrease in
integrated intensity suggests the effective potential approaches a harmonic
curve at high temperatures. We therefore use the following DW potential form:

$V(x)=Ax^{2}+B\mathcal{G}(x,0,\sigma)$ (2a)
$\mathcal{G}(x,\mu,\sigma)=(2\pi\sigma^{2})^{\text{{-}}\frac{1}{2}}e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}$
(2b)

where $\mathcal{G}(x,\mu,\sigma)$ is a Gaussian barrier with $A,B,\sigma\geq
0$.

Given a set of eigenstates, $\psi_{n}(x)\equiv\left\langle
x\middle|n\right\rangle$, the temperature dependent probability density of the
particle is given by

$\mathcal{P}(x)=\sum_{n}\frac{e^{-E_{n}\beta}}{Z_{T}}\left|\psi_{n}\left(x\right)\right|^{2}~{},$
(3)

with $E_{n}$ the eigenvalue and $Z_{T}=\sum_{n}e^{-E_{n}\beta}$ the canonical
partition function. Isolating the effect of lattice dynamics, the 2nd-order
inelastic scattering spectra were calculated from the four-point correlation
function for particle position (the subscript $2$ denotes 2nd-order scattering
- see Sec. S3 in the SM for full derivation):

$\begin{split}\mathcal{S}_{2}(\omega)&\propto\int\text{{d}}t~{}e^{-i\omega
t}\left\langle x(t)x(t)x(0)x(0)\right\rangle_{T}\\\
&\propto\sum_{n}\sum_{m}\frac{e^{-E_{n}\beta}}{Z_{T}}\left|\left\langle
m\middle|x^{2}\middle|n\right\rangle\right|^{2}\delta\left(\omega_{nm}-\omega\right)~{},\end{split}$
(4)

with the temperature-dependent integrated intensity given by:

$I=\int\mathcal{S}_{2}(\omega)~{}\text{{d}}\omega~{},$ (5)

where $\omega_{nm}\equiv(E_{n}-E_{m})/\hbar$, $\beta=(k_{B}T)^{-1}$ and
$\hbar$ and $k_{B}$ are the Planck and Boltzmann constants. The same
calculation was performed for a particle in a perfect harmonic potential
($B=0$) for comparison.

Figure 2(a) shows the probability density calculated at various temperatures
for the case of a DW potential. At low temperatures the particle is confined
to one of the wells, with a sharp Gaussian probability density around the
well’s minima ($x=\pm W$) and almost none at the barrier. As temperature
rises, the probability density broadens symmetrically, increasing in the
barrier. As expected, inside the harmonic potential the particle is localized
at the bottom of the well ($x=0$), and the probability density broadens with
increasing temperature (see Fig. S6 in the SM).

Figures 2(b) and 2(c) show the simulated 2nd-order Raman spectra at
$T=500~{}\text{{Ha}}$ and the temperature dependent integrated intensity,
calculated by Eq. (4) and (5), respectively, for the DW (blue) and parabolic
(red) potential. The effect of the potential barrier is apparent - an excess
in scattering intensity compared to the harmonic case (green shading).
Moreover, the largest relative difference arises at low temperatures. The full
simulated spectra are depicted in Fig. S5 in the SM.

Our harmonic spectra agree with the analytical solution, showing a single
sharp peak at the overtone frequency (see Sec. S3 in the SM). Comparison of
the harmonic and DW spectra show the excess intensity stems from low-frequency
features in the DW simulation, absent in the harmonic case. These originate in
transitions between nearly degenerate localized states that are forbidden in
the harmonic case. This may be understood as spontaneous symmetry breaking,
where localization of the atomic states relaxes the harmonic selection rules
for the matrix element $\left\langle m\middle|x^{2}\middle|n\right\rangle$.
[45] This effect persists at high temperatures, far above the barrier height
(Fig. S6 in the SM). However, had that been the whole story, we would expect
the integrated intensity to monotonically increase with temperature as shown
in Fig. 2(b). Figure 2(d) shows a monotonic dependence of the excess intensity
at $T=500~{}\text{{Ha}}$ on the barrier height. Thus, a decreasing barrier can
explain the observed temperature dependence. This explains why other static-
potential theories with symmetry breaking [46, 47, 48, 49] are insufficient to
explain the observed temperature dependence.

To apply and test these lessons on the experimental spectra we propose a
phenomenological model for 2nd-order Raman scattering, with the effect of a DW
potential realized by a low-frequency feature, corresponding to symmetry
breaking fluctuations. The effective temperature dependence of the potential
barrier is introduced through a fitting parameter $\eta(T)$ that allows the
system to gradually reduce to harmonic dynamics, restoring the cubic selection
rules at high temperatures.

### Description of The Model

As in the numerical simulation, we conserve the cubic point group constraints
and assume the crystal is 1st-order Raman inactive, such that the leading term
becomes two-phonon Raman scattering. Formally, 2nd-order Raman scattering
involves only transitions that conserve momentum, i.e. any combination of two
modes, at $\bm{q}$ and $\bm{q}^{\prime}=-\bm{q}\equiv\bm{\bar{q}}$. [42] As
the Fourier transform of a four-point correlation function, the 2nd-order
Stokes cross-section may be approximated as the convolution: [50]

$\begin{split}\mathcal{S}_{2}(\omega,0)\propto\sum_{j,j^{\prime},\bm{q}}\chi_{j,j^{\prime}}^{\bm{q}}&\int\text{{d}}\Omega~{}\mathcal{J}_{j}(\Omega,\bm{q})[n(\Omega,T)+1]\\\
\times~{}&\mathcal{J}_{j^{\prime}}(\omega-\Omega,\bm{\bar{q}})[n(\omega-\Omega,T)+1]\\\
=\sum_{j,j^{\prime},\bm{q}}\chi_{j,j^{\prime}}^{\bm{q}}&~{}\mathcal{J}_{j}(T,\omega,\bm{q})\ast\mathcal{J}_{j^{\prime}}(T,\omega,\bm{\bar{q}})~{},\end{split}$
(6)

where $\chi_{j,j^{\prime}}^{\bm{q}}$ are the 2nd-order susceptibility mode
derivatives, $n(\omega,T)$ is the Bose-Einstein occupation factor (see Eq.
(S8) in the SM) and we use the shorthand $f(T,\omega,\bm{q})\equiv
f(\omega,\bm{q})[n(\omega,T)+1]$. The spectral function
$\mathcal{J}_{j}(\omega,\bm{q})$ is defined as:

$\begin{split}\mathcal{J}_{j}(\omega,\bm{q})=[n(\omega,&T)+1]^{-1}\\\
&\times\int\text{{d}}t~{}e^{-i\omega t}\left\langle
U_{j}(\bm{q},t)U_{j}(\bm{q},0)\right\rangle_{T}~{},\end{split}$ (7)

the Fourier transform of the thermally averaged
($\left\langle\cdots\right\rangle_{T}$) auto-correlation function of atomic
displacement $U_{j}(\bm{q},t)$ for mode $j$. The spectral function for a
crystal of N${}_{\text{{uc}}}$ unit cells must satisfy: [51]

$\int_{0}^{\infty}\text{{d}}\Omega~{}\mathcal{J}_{j}(\Omega,\bm{q})=1$ (8a)
$\frac{1}{\text{{N}}_{\text{{uc}}}}\sum_{j,\bm{q}}\mathcal{J}_{j}(\omega,\bm{q})=g(\omega)$
(8b)

where $g(\omega)$ is the vibrational density of states (vDOS).

To introduce the spontaneous symmetry breaking created by the DW potential, as
indicated by simulation, we invoke a low-frequency feature (LFF),
$\mathcal{D}(\omega,\bm{q})$, in the spectral function. The LFF is not
included in $g(T,\omega)$ and is completely decoupled from phononic vibrations
in the sense that there is no energy transfer between the two modalities. This
LFF can represent a low-frequency vibration that breaks the global symmetry, a
localized fluctuation, or act as an effective description of multiple
oscillations. [52, 53, 54, 55] A similar approach was introduced by Fontana
and Lambert [23] for the Raman spectra of BaTiO3, by Safran et al. [56] for
magnetic phase transitions, and by Dultz and others [57, 58, 59, 60] for
disorder-induced light scattering. We thus divide the spectral function to:

$\sum_{j,\bm{q}}\mathcal{J}_{j}(\omega,\bm{q})=\eta\sum_{\bm{q}}\mathcal{D}(\omega,\bm{q})+(1-\eta)\sum_{j,\bm{q}}\mathcal{P}_{j}(\omega,\bm{q})~{},$
(9)

where $\mathcal{P}_{j}(\omega,\bm{q})$ represents the phononic part in the
spectral function, and $0\leq\eta\leq 1$ is the LFF component fraction. The
LFF component is normalized by

$\frac{1}{\text{{N}}_{\text{{uc}}}}\int_{0}^{\infty}\text{{d}}\Omega~{}\sum_{\bm{q}}\mathcal{D}(\Omega,\bm{q})=3N~{},$
(10)

where $N$ is the number of atoms per unit cell, such that conditions (8) for
the full spectral function hold.

We further assume that $\mathcal{D}(\omega,\bm{q})$ is a separable function:

$\mathcal{D}(\omega,\bm{q})=D(\omega)C(\bm{q})~{}.$ (11)

Neutron scattering and statistical mechanics indicate that $C(\bm{q})$ has a
Lorentzian line-shape, centered at $\bm{q}_{0}$. [61, 62, 63, 64, 55]
According to the Orstein-Zernike theory in one dimension [65], it would have
the form:

$\begin{split}C(q)&=\frac{T/T_{c}}{a^{2}(\bm{q}-\bm{q_{0}})^{2}+(T-T_{c})/T_{c}}\\\
&=\frac{\xi^{2}(\xi^{2}-1)^{-1}}{a^{2}(\bm{q}-\bm{q_{0}})^{2}+(\xi^{2}-1)^{-1}}~{},\end{split}$
(12)

where $a$ is the lattice constant and $\xi$ is the temperature-dependent
correlation length in units of unit-cells, defined as $\xi^{2}=T/(T-T_{c})\geq
1$. As temperature approaches the phase transition ($T\rightarrow T_{c}$), the
correlation length $\xi$ diverges, and the fluctuation leads the phase
transition into the ordered, lower temperature phase. $\bm{q}_{0}$ is the
high-temperature phase fluctuation wave-vector, and it defines the symmetry
breaking across the two phases. [66, 67] For example, in ferro-distortive
phase transition $\bm{q}_{0}=0$, [24] whereas in anti-ferro-distortive phase
transition $\bm{q}_{0}$ would be the Brillouin zone edge. [5, 68, 69]

Plugging Eq. (9) and (11) back into Eq. (6) and approximating
$\chi_{j,j^{\prime}}^{\bm{q}}\equiv 1$, we find the cross-section is composed
of three parts:

$\begin{split}\mathcal{S}_{2}(\omega,0)\propto\eta^{2}\left[\sum_{\bm{q}}C(\bm{q})C(\bm{\bar{q}})\right]D(T,\omega)\ast
D(T,\omega)\\\
+2\eta(1-\eta)\left[\sum_{j,\bm{q}}C(\bm{q})\mathcal{P}_{j}(T,\omega,\bm{\bar{q}})\right]\ast
D(T,\omega)&\\\
+(1-\eta)^{2}\sum_{j,j^{\prime},\bm{q}}\mathcal{P}_{j}(T,\omega,\bm{q})\ast\mathcal{P}_{j^{\prime}}(T,\omega,\bm{\bar{q}})&~{},\end{split}$
(13)

the first represents 2-LFF scattering, i.e., the creation of two LFF
excitations in the case of Stokes scattering. The second involves 1-LFF and
1-phonon excitation. The final third term is equivalent to the standard
expression for 2-phonon scattering [42]. Although detrimental for spectral
accuracy, the removal of $\chi_{j,j^{\prime}}^{\bm{q}}$-dependency is not as
crude as might first appear; selection rules for 2nd-order scattering are
usually so lax as to allow most phonon pairs to participate, even in high
symmetry structures, [22] and it has long been known that the 2-phonon vDOS
can provide a rough estimation of the 2nd-order Raman spectrum. [70] Most
importantly, as we demonstrate below, this simplified version is sufficient to
capture and explain the non-monotonic temperature dependence of the integrated
intensity.

Assuming the frequency of $\mathcal{D}(T,\omega,\bm{q})$ is much lower than
phonon frequencies, we define the vDOS as
$\text{{N}}_{\text{{uc}}}^{-1}\sum_{j,\bm{q}}\mathcal{P}_{j}(T,\omega,\bm{q})=g(T,\omega)$.
Simplifying further by setting $C(\bm{q})\equiv 1$, the cross-section becomes

$\begin{split}\mathcal{S}_{2}(\omega,0)\propto\eta^{2}~{}D(T,\omega)\ast
D(T,\omega&)\\\ +2\eta(1-\eta)~{}g(T,\omega)\ast D&(T,\omega)\\\
+(1-\eta)^{2}~{}g(T,\omega&)\ast g(T,\omega)~{},\end{split}$ (14)

where we used Eq. (8b) to find that $\mathcal{S}_{2}(\omega,0)$ is given by
the convolutions of $g(T,\omega)$ with itself and with $D(T,\omega)$.

Figure 3:  Fit results of Eq. (14) to the data of BaTiO3, SrTiO3 and CsPbBr3
at three selected temperatures. Blue and red traces are the data and model,
respectively.

### Fitting to the Model

Assuming $g(T,\omega)$ is known and $D(T,\omega)$ is a damped Lorentz
oscillator, [29] Eq. (14) has 4 remaining fitting parameters: (1) $A$ \-
global proportionality factor of $\mathcal{S}_{2}(\omega,0)$, (2) $\eta$ \-
weight of the LFF in the spectral function, (3) $\omega_{D}$ \- frequency of
the LFF, (4) $\Gamma_{D}$ \- damping coefficient of the LFF.

Figure 3 shows the fit results for BaTiO3, SrTiO3 and CsPbBr3 at selected
temperatures. We used experimental densities of states [71, 25, 5] from
inelastic neutron scattering at 300 K to calculate $g(T,\omega)$. The fitted
function $D(T,\omega)$ and $g(T,\omega)$ were normalized according to Eq.(8)
and (10).

We begin the fitting process by matching the highest temperature integrated
intensity to the model with $\eta=0$, i.e., to conventional 2-phonon
scattering. This procedure fixes the proportionality factor, which is set
throughout all temperatures. Next, we use the Lorentz parameters $\omega_{D}$,
$\Gamma_{D}$, and the LFF weight component $\eta$, to fit the integrated
intensity as function of temperature. As the fitted plots in Fig. 3 show, by
adding $D(T,\omega)$ to the spectral function we are able to reproduce the
non-monotonic temperature evolution of the Raman spectra. Discrepancies in the
spectral shape are the result of our simplifying approximations
$C(\bm{q}),\chi_{j,j^{\prime}}^{\bm{q}}\equiv 1$, and complete decoupling
between phonons and the LFF. [72, 73, 74]

We emphasize that it is difficult to attach physical meaning to the fitted
parameters without additional constraints on either $D(T,\omega)$ or $\eta$.
The final values and the integrated intensities are depicted in Fig. S7 and S7
in the SM. The number of parameters may be reduced further by establishing the
origin of $D(\omega)$ and independently estimating $\omega_{D}$ and
$\Gamma_{D}$ or the relation $\Gamma_{D}=\Gamma(\omega_{D})$.

Nonetheless, close examination of the fitting process revealed the possible
different mechanisms governing the temperature dependence. A slow increase in
intensity correlates with an increase in the frequency of the LFF
($\omega_{D}$), while a decrease in intensity correlates with a decrease in
LFF contributions ($\eta$). Our interpretation of the Raman spectra as 2nd-
order scattering enhanced by a LFF also resolves some of the difficulties with
competing interpretations mentioned above. It reproduces the non-monotonic
temperature dependence without challenging the selection rules derived for the
cubic average structure. Moreover, because its low frequency and localization
in space, the LFF should not couple efficiently to electrons. Therefore, it
remains below the frequency range of THz-absorption measurements, and does not
affect the electronic band structure, allowing for a sharp absorption edge.

Our model also offers insight into another aspect of the different perovskite
Raman spectra. There is a striking difference between the sharp central peak
of BaTiO3 and the shallow central feature of SrTiO3 and CsPbBr3. It was
already suggested that the central feature in CsPbBr3 is related to scattering
by polar fluctuation. [29, 28]. However, our calculations show that the
central feature can be reproduced without the 2-LFF term in SrTiO3 and
CsPbBr3, but not in BaTiO3. This suggests the central peak of BaTiO3 is indeed
directly induced by the low-frequency anharmonic motion leading the phase
transition and stabilizing the cubic phase. As temperature approaches $T_{c}$,
the correlation length of the symmetry-breaking motion increases, leading to
an effective tetragonal local structure even before the phase transition. In
both CsPbBr3 and SrTiO3 the phase transition is lead by vibrations with
$\bm{q}$-vectors at the Brillouin zone edge [5, 68, 69], but since BaTiO3 goes
into a ferroelectric phase, its transition is lead by a $\Gamma$-point
vibration. [24] It is therefore possible that as the correlation length grows,
cubic selection rules in BaTiO3 relax, leading to 1st-order scattering and a
sharp central peak.

## Conclusions

We presented numerical simulations of a 1D quantum particle, showing the
effect of a DW potential on the Raman scattering of disordered crystals. The
DW induces low-frequency features interpreted as transitions between potential
wells, causing an excess in scattering intensity, which can explain the
abnormally high Raman cross-section in cubic perovskites. The simulation
results link the decrease in scattering intensity in higher temperatures to a
temperature-dependent potential surface, which evolves from a multi-well to
harmonic-like potential. Motivated by this insight, we presented a
phenomenological 2nd-order Raman scattering model that conserves the cubic
symmetry of the crystal structure. By introducing a low-frequency feature and
incorporating it into the 2nd-order scattering formalism, we successfully
reproduced the broad Raman spectra and non-monotonic temperature dependence.
The existence of this low-frequency feature is supported by various
observations, such as over-damped modes and critical inelastic neutron
scattering, [73, 75, 2, 76, 77, 5] central peak in Raman scattering, [23, 29,
28, 11, 17, 16, 41] soft modes, [78, 79] short phonon coherence length [80,
81, 82] and a high dielectric function at low-frequencies. [83, 84, 74, 1]

Our interpretation is disencumbered by contradictions with Raman selection
rules or complementary measurements. The model similarly applies to oxide and
halide perovskites, emphasizing the similarities between the crystal families
and their interaction with light. It can be applied to other dynamically-
disordered crystal phases, providing better intuition of the effect of
structural dynamics on the electronic, optical, and mechanical properties of
functional materials.

### Acknowledgements

The authors would like to thank Dr. Iddo Pinkas for help in designing the
experimental setup, and Dr. Lior Segev (WIS) for developing the Raman
software. O.Y. acknowledges funding from: ISF(1861/17), BSF (grant
No.2016650), ERC (850041 - ANHARMONIC).

##  Methods

### Raman Scattering

Raman scattering measurements were conducted in a home-built back-scattering
system using a 2.54 eV (for BaTiO3 and SrTiO3) CW sapphire pumped-diode laser
(Coherent Inc., USA) and a 1.58 eV CW pump-diode laser (Toptica Inc., USA).
The samples were heated in a closed temperature control system (Linkam
Scientific) under continues Nitrogen flow. Measures taken to isolate the
effect of temperature on the intensity involve: maintaining constant
measurement conditions, careful refocusing of the laser before every
measurement and repeated measurement cycles (heating and cooling) across the
full temperature range and phase transitions. The Stokes integrated intensity
was calculated by numerically integrating over the experimental spectra in the
Stokes scattering side ($\omega>0$).

Supplemental Material
The Disorder Origin of Raman Scattering In Perovskites Single Crystals

##  S1 Materials

Single crystals of BaTiO3 and SrTiO3 with (001) orientation were purchased
from MTI cooporation (Richmond, CA, USA) and CrystTech GmbH (Berlin, Germany),
respectively. A single crystal of CsPbBr3 was synthesized according to the
procedure described in Ref. [85] and characterized by optical microscopy,
x-ray diffraction, and Raman scattering (data not shown).

##  S2 Raman Scattering

Raman measurements were preformed in a system designed for polarization
orientation (PO) Raman measurements. The full scheme is described thoroughly
in Ref. [10, 86, 11, 50, 12] and the chosen incident laser lines are described
in the Methods section in the main text.

The unpolarized Raman spectra are obtained by averaging the parallel and
perpendicular spectra at two perpendicular directions. This procedure is
equivalent to measuring with an unpolarized laser source and no analyzer. The
advantage of this process is that we require a scrambler to avoid the inherent
polarization of the laser source.

The polarization dependent intensity for a given mode is given by the
following equation: [70]

$\mathcal{S}\propto|\hat{e}_{s}\cdot\bm{J}\cdot\bm{R}\cdot\bm{J}\cdot\hat{e}_{i}|^{2}$
(S1)

Where $\hat{e}_{i}$ and $\hat{e}_{s}$ are the polarization vectors of the
incident and scattered light, respectively. $\bm{R}$ is the Raman tensor, and
$\bm{J}$ is the Jones matrix, [87, 88] introduced to account for birefringence
effects due to anisotropy in the crystal structure. [89, 90, 91] $\hat{e}_{i}$
and $\hat{e}_{s}$ are either parallel or perpendicular to one another,
depending on the measurement configuration.

In principle, the Raman tensor is a $3\times 3$ matrix, constrained by
symmetry considerations. The shape of the Raman tensor depends on the symmetry
of the crystal, symmetry of the mode and the crystal face measured. To
generalize our claims, we consider scattering from a general Raman tensor,
unconstrained by symmetry. Rotation of the Raman tensor to different
crystallographic orientations would mix the diagonal and off-diagonal elements
of the tensor. Therefore, we use the most general real-valued tensor:

$\bm{R}=\begin{pmatrix}a&c\\\ d&b\end{pmatrix}$ (S2)

Our polarization manipulation is confined to the $xy$ plane of the laboratory
frame. Therefore, we are blind to the $z$ components of $\bm{R}$ and may
consider it as a $2\times 2$ tensor. In our laboratory frame, the polarization
vectors are given by:

$\hat{e}_{i}=\left(\begin{array}[]{c}cos(\theta)\\\
sin(\theta)\end{array}\right)~{},~{}\hat{e}_{S,||}^{T}=\left(\begin{array}[]{c}cos(\theta)\\\
sin(\theta)\end{array}\right)~{},~{}\hat{e}_{S,\perp}^{T}=\left(\begin{array}[]{c}-sin(\theta)\\\
cos(\theta)\end{array}\right)$ (S3)

and the Jones-matrix is given by:

$\bm{J}=\left(\begin{array}[]{cc}1&0\\\
0&e^{i\left|\phi\right|}\end{array}\right)$ (S4)

Plugging all definitions back to Eq. (S1) and solve, we find:

$\mathcal{S}_{\parallel}(\theta)\propto\left|a\cos^{2}(\theta)+(c+d)e^{i\left|\phi\right|}\sin(\theta)\cos(\theta)+be^{2i\left|\phi\right|}\sin^{2}(\theta)\right|^{2}$
(S5a)

$\mathcal{S}_{\perp}(\theta)\propto\left|de^{i\left|\phi\right|}\cos^{2}(\theta)-\left(a-be^{2i\left|\phi\right|}\right)\sin(\theta)\cos(\theta)-ce^{i\left|\phi\right|}\sin^{2}(\theta)\right|^{2}$
(S5b)

where $\mathcal{S}_{\parallel}(\theta)$ and $\mathcal{S}_{\perp}(\theta)$ are
the intensity in the parallel and perpendicular configurations, respectively.
The unpolarized spectra are obtained by summing:

$\begin{split}\mathcal{S}_{\text{{unpol}}}&=\mathcal{S}_{\parallel}(\theta)+\mathcal{S}_{\perp}(\theta)+\mathcal{S}_{\parallel}\left(\theta+\frac{\pi}{2}\right)+\mathcal{S}_{\perp}\left(\theta+\frac{\pi}{2}\right)\\\
&=a^{2}+b^{2}+c^{2}+d^{2}\\\
&=\frac{1}{\pi}\int_{0}^{2\pi}\mathcal{S}_{\parallel}(\theta)+\mathcal{S}_{\perp}(\theta)~{}\text{{d}}\theta\end{split}$
(S6)

which is independent of $\theta$. The generality of the Raman tensor and the
restriction to the laboratory frame guarantee the result is applicable to all
modes in the spectrum, and all orders of scattering. Therefore, we can apply
Eq. (S6) to the PO Raman spectra regardless of the choice of $\theta$. To
properly follow the intensity evolution with temperature, $\theta$ was kept
constant throughout the temperature cycles, and all spectra were normalized to
a single global value.

Figure S1: All Temperature-dependent unpolarized Raman scattering spectra of
BaTiO3, SrTiO3 and CsPbBr3 in the cubic phase. The spectra of BaTiO3 and
SrTiO3 are divided to two scattering regimes, showing the non-monotonic
temperature dependence.

Figure S1 presents the unpolarized Raman spectra of BaTiO3, SrTiO3 and
CsPbBr3. The data in Fig. 1 in the main text were taken from this data set.
The spectra of BaTiO3 and SrTiO3 are divided to two scattering regimes, as
described in the main text.

###  S2.a  SrTiO3 data below 300 K

Unlike BaTiO3 and CsPbBr3, the phase transition temperature of SrTiO3 is below
room temperature ($T_{c}=$ 105 K). Therefore, to measure the Raman spectra of
SrTiO3 in the range of $T_{c}\leq T\leq$300 K the measurement system had to be
modified. The SrTiO3 crystal was loaded into liquid N2 cooled optical-cryostat
(Janis Inc., USA), set up in the same experimental system, instead of the
closed temperature control system (Linkam Scientific). The same spot was
measured for both data sets, below and above 300 K.

Figure S2 shows the temperature dependent unpolarized Raman spectra of SrTiO3
between the tetragonal-cubic phase transition temperature and room
temperature. The spectra show the same trends with temperature as shown in
Fig. S1. Comparing the spectra below (100 K, black trace) and above (110 K,
blue trace) the phase transition, we see a very subtle effect on the Raman
spectra. Small peaks in the spectrum of the tetragonal phase, assigned to 1st-
order Raman scattering [92, 93] or localized modes [94], disappear as we cross
the phase transition to the cubic phase.

Figure S2: Temperature dependent Raman scattering spectra of cubic SrTiO3
below room temperature. The spectra show the same trends with temperature as
shown in Fig. S1 below room temperature down to the phase transition
temperature ($T_{c}=$105 K). Raman spectrum at the tetragonal phase ($T=$ 100
K) is in black, showing the same broad features appearing in the cubic phase,
together with small peaks assigned as 1st-order Raman scattering peaks. The
intensity of the 2nd-order scattering is as high as the the 1st-order
scattering in the tetragonal phase.

###  S2.b  BaTiO3 Central Peak in PO

When exciting BaTiO3 single crystal in our PO Raman setup, we can selectively
observe only the central peak. This is done when the excitation polarization
is parallel to one of the crystallographic axis, and collection of light
scattered perpendicular to the incident polarization ($Z(XY)Z$ or $Z(YX)Z$ in
Porto notation [95]). This scheme allows for observation of modes with non-
vanishing off-diagonal components in the Raman tensor. Fortunately, this
configuration emphasizes the central peak in BaTiO3 by rejecting many modes
around it.

Figure S3: Temperature dependent polarized Raman scattering spectra of cubic
BaTiO3 showing mostly the central peak. Incident polarization is along one of
the crystallographic axes, and scattered polarization is perpendicular
($Z(XY)Z$ or $Z(YX)Z$ in Porto notation). The spectra are divided to the two
scattering regimes as described in the main text.

Conventionally, the central peak is related to a long-amplitude relaxational
motion with no restoring fore. [29, 72, 41] Therefore, to extract the
temperature dependence of the central peak in BaTiO3, the spectra were fitted
to a Debye relaxor and a sum of damped Lorentz oscillators multiplied by the
Bose-Einstein occupation number:

$\begin{split}S(\omega)=[n(\omega,&T)+1]\frac{c_{0}|\omega|\Gamma_{0}}{\omega^{2}+\Gamma_{0}^{2}}~{}~{}+\\\
[&n(\omega,T)+1]\sum_{i}\frac{c_{i}\omega\Gamma_{i}^{3}}{\omega^{2}\Gamma_{i}^{2}+\left(\omega^{2}-\Omega_{i}^{2}\right)^{2}}\end{split}$
(S7)

where subscripts $0/i$ stand for the Debye and $i^{th}$ Lorentz oscillator,
$\omega$ and $\Omega_{i}$ are the probing and mode-central frequency
respectively, $\Gamma_{0/i}$ is the damping constant inversely proportional to
the lifetime of the fluctuation, and $c_{0/i}$ is a unit-less fitting
parameter. The Bose-Einstein occupation number as function of $\omega$ and the
absolute temperature $T$ is given by:

$n(\omega,T)=(e^{\beta\hbar\omega}-1)^{-1}$ (S8)

where $\beta=(k_{B}T)^{-1}$, $k_{B}$ and $\hbar$ are the Boltzmann and reduced
Planck’s constants. Damped Lorentz oscillators were used to capture the rest
of the spectral features, without assuming any more meaning to the fit
results. The integrated intensity of the central peak was extracted by
integrating the Debye relaxor term as function of frequency.

Figure S3 shows the temperature dependent polarized Raman spectra of BaTiO3 in
this configuration, showing mostly the central peak. The spectra are divided
into the two scattering regimes as in the Fig. 1 in the main text. The
decreasing intensity of the central peak is clearly evident.

###  S2.c Comparing to 1- and 2-phonon scattering

To compare the temperature dependence of the integrated intensity of BaTiO3,
SrTiO3 and CsPbBr3 to the expected dependencies from 1st\- and 2nd-order Raman
scattering, we have to approximate the expected dependencies using the
available data.

The structure factor in 1st-order Raman scattering at a given temperature has
the form:

$\mathcal{S}_{1}(\omega,0)_{T_{0}}\propto\sum_{j}\chi_{j}\mathcal{J}_{j}(\omega,0)[n(\omega,T_{0})+1]$
(S9)

where $\mathcal{S}_{m}(\omega,\bm{q})$ is the structure factor of
$m^{\text{th}}$ order Raman scattering at angular-frequency $\omega$ and
scattering wave-vector (and crystal momentum) $\bm{q}=0$, due to momentum
conservation. $\chi_{j}$ is the Raman tensor of mode $j$ and
$\mathcal{J}_{j}(\omega,\bm{q})$ is the spectral function, defined in Eq. (7)
in the main text. The temperature dependence of 1st-order Raman follows the
Bose-Einstein distribution, as described in Eq. (S8). Conventionally, it is
assumed that the spectral function does not depend on temperature. Therefore,
the expected temperature dependent integrated intensity can be approximated
by:

$I(T)=\int_{0}^{\infty}\text{{d}}\omega~{}\mathcal{S}_{1}(\omega,0)_{T_{0}}\frac{[n(T,\omega)+1]}{[n(T_{0},\omega)+1]}$
(S10)

The structure factor in 2nd-order Raman scattering at a given temperature was
taken from Eq. (14) in the main text, with $\eta=0$:

$\mathcal{S}_{2}(\omega,0)_{T_{0}}=[n(T_{0},\omega)+1]g(\omega)\ast[n(T_{0},\omega)+1]g(\omega)$
(S11)

where we assume a temperature independent density of states. Therefore, the
expected temperature dependent integrated intensity was estimated by
calculating $\mathcal{S}_{2}(\omega,0)_{T}$ for any given temperature and
integrating:

$I(T)=\int_{0}^{\infty}\text{{d}}\omega~{}\mathcal{S}_{2}(\omega,0)_{T}$ (S12)

Figure S4 shows the temperature dependent Stokes integrated intensity of the
Raman spectra of BaTiO3, SrTiO3 and CsPbBr3, compared to the calculated
expected dependencies according to Eq. (S10) and (S11). The calculated
dependencies were normalized to cross the maximum intensity point. Clearly,
the experimental dependencies of BaTiO3 and CsPbBr3 do not follow the expected
behavior. In the case of SrTiO3, in the range of $T_{c}<T\leq T_{m}$ the
integrated intensity increases with temperature like 1st-order Raman
scattering, which is forbidden by the selection rules for the cubic perovskite
structure. Moreover, the decreasing integrated intensity cannot be reconciled
with the calculated intensities, which are monotonic with temperature.

Figure S4: Temperature-dependent Stokes integrated intensity (circles) of
BaTiO3, SrTiO3 and CsPbBr3 in the cubic phase compared with the expected
integrated intensity from 1-phonon (dense-dash) and 2-phonon (wide-dash)
scattering, normalized to cross the maximum intensity point. The integrated
intensity of BaTiO3 and CsPbBr3 do not follow any of the calculated
dependencies. The data of SrTiO3 does follow the 1-phonon line below 900 K,
which is violation of the Raman selection rules for the crystal structure.

##  S3 Numerical Simulation

To test the feasibility of our hypothesis, that a double-well (DW) potential
could give rise to a non-monotonic trend in Raman scattering, we numerically
solved the time-independent Schrödinger equation,

$\left(\frac{-\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}+V(x)\right)\psi(x)=E\psi(x)$ (S13)

with $\hbar$ the reduced Planck constant and $m$ the particle mass. Because
the solution is numerical we are able to solve for any random potential
$V(x)$. We used MATLAB software and a 1D grid of 2401 points, yielding 2401
eigenstates and corresponding eigenvalues. Specifically, Eq. (S13) was solved
for $m=\hbar=1$, and

$V_{\text{{Har}}}(x)=\frac{1}{2}m\omega^{2}x^{2}$ (S14)

for a perfect harmonic potential, and

$V_{\text{{DW}}}(x)=V_{\text{{Har}}}(x)+B\mathcal{G}(x,0,1)$ (S15)

for the DW potential. $\mathcal{G}(x,0,1)$ is a Gaussian barrier centered at
$x=0$ with a full-with-half-maximum of 1. The $x$ range for the solution was
chosen in an iterative process, making sure that at the highest temperature
$x_{\text{{max}}}>>\Delta x$ and the probability density to find the particle
decays to zero at $\pm x_{\text{{max}}}$. The finite nature of the calculation
necessarily gives a maximum value for the potential surface, and therefore not
all calculated states are bound. This created non-physical solutions for high
enough eigenvalues, but these were verified to have negligible contributions
in all pertinent temperatures. Moreover, to avoid additional non-physical
errors due to the numerical nature of the simulation, the solution for
$V_{\text{{Har}}}$ was compared to the analytical solution by evaluating:

$\frac{E_{n}}{\omega(n+1/2)}=\hbar=1~{}(\pm 0.1\%)$ (S16)

where $E_{n}$ is the $n^{th}$ eigenvalue and all eigenvalues deviated by more
than 0.1% were rejected.

The set of eigenvalues $\left\\{E_{n}\right\\}$ and eigenstates
$\left\\{\psi_{n}(x)\right\\}$ determines the temperature-dependent
probability density and Raman spectra. If the canonical ensemble density
operator is given by

$\rho(T)=\sum_{n}p_{n}(T)\left|\psi_{n}\rangle\langle\psi_{n}\right|=\sum_{n}\frac{e^{-E_{n}\beta}}{Z_{T}}\left|\psi_{n}\rangle\langle\psi_{n}\right|$
(S17)

with $Z_{T}=\sum_{n}e^{-E_{n}\beta}$ the canonical partition function, and
$\beta=(k_{B}T)^{-1}$, the probability density is given by

$\begin{split}\mathcal{P}(x)&=\text{{tr}}\left(\rho(T)|x\rangle\langle
x|\right)=\text{{tr}}\left(\sum_{n}p_{n}(T)|\psi_{n}\rangle\langle\psi_{n}|x\rangle\langle
x|\right)\\\ &=\int\text{{d}}x^{\prime}~{}\left\langle
x^{\prime}\left|\sum_{n}p_{n}(T)\right|\psi_{n}\right\rangle\langle\psi_{n}|x\rangle\langle
x|x^{\prime}\rangle\\\ &=\int\text{{d}}x^{\prime}~{}\left\langle
x^{\prime}\left|\sum_{n}p_{n}(T)\right|\psi_{n}\right\rangle\langle\psi_{n}|x\rangle\delta\left(x-x^{\prime}\right)\\\
&=\sum_{n}p_{n}(T)\langle
x|\psi_{n}\rangle\langle\psi_{n}|x\rangle=\sum_{n}\frac{e^{-E_{n}\beta}}{Z_{T}}|\psi_{n}(x)|^{2}\end{split}$
(S18)

To calculate the Raman spectra we start from [96]:

$\mathcal{S}(\omega)\propto\int_{-\infty}^{\infty}\text{{d}}\tau~{}e^{-i\omega\tau}\langle\chi(\tau)^{*}\chi(0)\rangle_{T}$
(S19)

with $\chi(\tau)$ the time-dependent polarizability operator and
$\langle\cdot\rangle_{T}$ denoting a thermal average. We have no information
about the polarizability, but assume its time-dependence originates in the
ionic motion $x(t)$. We therefore Taylor expand the polarizability in atomic
displacement $x$:

$\chi(x)=\chi_{0}+\left(\frac{\text{{d}}\chi}{\text{{d}}x}\right)_{x=0}x+\left(\frac{\text{{d}}^{2}\chi}{\text{{d}}x^{2}}\right)_{x=0}x^{2}+\cdots.$
(S20)

Plugging the expansion back into Eq. (S19) we can isolate any scattering order
we desire. Due to the selection rules for cubic perovskite structure, we
isolate the 2nd-order contributions, to get:

$\mathcal{S}_{2}(\omega)\propto\int\text{{d}}t~{}e^{-i\omega t}\left\langle
x(t)x(t)x(0)x(0)\right\rangle_{T}$ (S21)

where the auto-correlation is now for displacements instead of
polarizabilities. Although our solutions are stationary, frequency-dependent
spectra may be obtained through the constraint of energy conservation and the
Fourier decomposition of a delta-function. Using the Heisenberg representation
for the location operator,
$\hat{x}\left(\tau\right)=e^{i\frac{H\tau}{\hbar}}\hat{x}e^{-i\frac{H\tau}{\hbar}}$,
Eq. (S21) becomes:

Figure S5: Simulated inelastic light scattering (Raman) spectra from a
particle in a an harmonic (top panels) and a double-well (bottom panels)
potential. a) All calculated spectra. b) Normalized spectra at
$T=10~{}\text{{Ha}}$ and $T=500~{}\text{{Ha}}$ showing the difference in
dominating features due to thermal population.

$\begin{split}\int\text{{d}}\tau e^{-i\omega\tau}&\left\langle
x(\tau)x(\tau)x(0)x(0)\right\rangle_{T}=\\\ &=\int\text{{d}}\tau
e^{-i\omega\tau}\sum_{n}p_{n}(T)\left\langle
n\left|U^{\dagger}(\tau)\hat{x}U(\tau)U^{\dagger}(\tau)\hat{x}U(\tau)\hat{x}\hat{x}\right|n\right\rangle\\\
&=\int\text{{d}}\tau e^{-i\omega\tau}\sum_{n}p_{n}(T)\left\langle
n\left|U^{\dagger}(\tau)\hat{x}\hat{x}U(\tau)\hat{x}\hat{x}\right|n\right\rangle\\\
&=\int\text{{d}}\tau e^{-i\omega\tau}\sum_{n}p_{n}(T)\left\langle
n\left|e^{i\frac{H\tau}{\hbar}}\hat{x}\left|l\right\rangle\left\langle
l\middle|\hat{x}\middle|m\right\rangle\left\langle
m\right|e^{-i\frac{H\tau}{\hbar}}\hat{x}\left|k\right\rangle\left\langle
k\right|\hat{x}\right|n\right\rangle\\\ &=\int\text{{d}}\tau
e^{-i\omega\tau}\sum_{n}p_{n}(T)\left\langle
n\left|e^{i\frac{E_{n}\tau}{\hbar}}\hat{x}\left|l\right\rangle\left\langle
l\middle|\hat{x}\middle|m\right\rangle\left\langle
m\right|e^{-i\frac{E_{m}\tau}{\hbar}}\hat{x}\left|k\right\rangle\left\langle
k\right|\hat{x}\right|n\right\rangle\\\ &=\sum_{n,m,l,k}p_{n}(T)\left\langle
n\middle|\hat{x}\middle|l\right\rangle\left\langle
l\middle|\hat{x}\middle|m\right\rangle\left\langle
m\middle|\hat{x}\middle|k\right\rangle\left\langle
k\middle|\hat{x}\middle|n\right\rangle\int\text{{d}}\tau
e^{i\left(\omega_{nm}-\omega\right)\tau}\\\
&=2\pi\sum_{n,m,l,k}p_{n}(T)\left\langle
n\middle|\hat{x}\middle|l\right\rangle\left\langle
l\middle|\hat{x}\middle|m\right\rangle\left\langle
m\middle|\hat{x}\middle|k\right\rangle\left\langle
k\middle|\hat{x}\middle|n\right\rangle\delta\left(\omega_{nm}-\omega\right)\\\
&=2\pi\sum_{n,m}p_{n}(T)\left\langle
n\middle|\hat{x}^{2}\middle|m\right\rangle\left\langle
m\middle|\hat{x}^{2}\middle|n\right\rangle\delta\left(\omega_{nm}-\omega\right)\end{split}$
(S22)

with $\omega_{nm}\equiv\frac{E_{n}-E_{m}}{\hbar}$.

The matrix elements can now be directly evaluated by

$\left\langle
n\middle|\hat{x}^{2}\middle|m\right\rangle=\int_{x_{\text{{min}}}}^{x_{\text{{max}}}}\text{{d}}x~{}\psi_{m}^{*}(x)\cdot
x^{2}\cdot\psi_{n}(x)$ (S23)

and the delta function is numerically approximated by a square function of a
single pixel in our $x$-axis.

Finally, the integrated intensities are given by

$I=\int_{0}^{\omega_{\text{{max}}}}\mathcal{S}_{2}(\omega)~{}\text{{d}}\omega.$
(S24)

Figure S6: a) Probability densities of finding the particle in a parabolic
potential as function of temperature showing the expected Guassian probability
($x_{\text{{max}}}=16$). b) The ratio between the probability density at the
barrier ($P(x=0)$) and the probability density at the bottom of the well
($P(W)$) under a double-well potential. The ratio is small at low
temperatures, when the particle is confined to the wells, and increases with
temperature as the probability density broadens. c) Integrated intensity of
the simulated 2nd-order Raman spectra as function of temperature, at higher
temperature range, under parabolic (red) and double-well (blue) potentials,
showing an increase in the excess intensity (green) with temperature.

Figure S5(a) shows the simulated 2nd-order Raman spectra as a function of
temperature for the harmonic (top panel) and DW (bottom panel) potential. The
Raman spectra of the harmonic potential exhibit a single peak at
$2\omega=20~{}cm^{-1}$ while the Raman spectra of the DW potential exhibit
many transitions around $2\omega$ as well as low frequency features. Figure
S5(b) shows the normalized simulated 2nd-order Raman spectra at
$T=10~{}\text{{Ha}}$ and $T=500~{}\text{{Ha}}$, emphasizing the dominant
features in each spectrum. Clearly, in the harmonic potential, the dominant
feature remains the $2\omega$ peak at all temperature range, while in the DW
potential the low frequency features dominate at low temperatures.

Figure S6(a) shows the probability density to find the particle in the
harmonic potential. As expected, the probability density is a Gaussian
distribution around the potential well which decays at the edges
($x_{\text{{max}}}=16$) and broadens with temperatures. The ratio between the
probability density at the top of the barrier and the bottom of the well
($r_{\mathcal{P}}=\mathcal{P}(0)/\mathcal{P}(W)$), as function of temperature,
is depicted in Fig. S6(b). $r_{\mathcal{P}}$ correlates very well with the DW
integrated intensity, as depicted in Fig. 2 in the main text. At low
temperatures, $r_{\mathcal{P}}$ is very low and $\mathcal{P}(0)$ is
negligible, and increases monotonically with temperature, approaching a
constant value. At the same low temperatures range, the DW integrated
intensity exhibits a sharp increase, and aligns parallel to the harmonic
integrated intensity with increasing temperature. This can be understood in
the context of spontaneous symmetry breaking: at low temperatures, when the
particle in confined to the wells, only transitions which are forbidden in the
harmonic case are activated and contribute to the integrated intensity. As
temperature increases, the particle feels an harmonic-like potential.
Harmonically allowed transitions are thermally activated and their relative
contribution increases rapidly, resulting in an harmonic-like integrated
intensity.

The monotonic increase in integrated intensity continues at higher
temperatures. Figure S6(c) shows the temperature dependent integrated
intensity of the simulated 2nd-order Raman spectra under an harmonic (red
trace) and DW (blue trace) potential and their difference (green trace), at a
higher temperatures. Clearly, the excess intensity increases monotonically
with temperature. The higher temperatures simulation was performed with a 1D
grid of 12001 points, to keep the nominal thermal contribution of high-energy
eigenstates negligible and a similar resolution in $x$.

Figure S7: a) Integrated intensity of SrTiO3 and BaTiO3 as function of
temperature compared to the integrated intensity from the fit. A good fit was
defined according to the match of the integrated intensities at all
temperatures. b) Fit result parameters of SrTiO3 (top) and BaTiO3 (bottom) as
function of temperature. The dashed lines are guides to the eye.

##  S4 Fit results

Figures S7 and S8 show the parameters from fitting the temperature dependent
Raman spectra of BaTiO3, SrTiO3 and CsPbBr3 (Fig. 1 in the main text) to Eq.
(14) in the main text. The fit was done by matching both the integrated
intensity and the spectral line-shape. The results reflect the chosen
constraints to the fit, where $\eta$ was kept constant in the range of
$T_{c}<T\leq T_{m}$, and was the only free fit parameter for $T>T_{m}$.

Figure S8: a) Integrated intensity of CsPbBr3 as function of temperature
compared to the integrated intensity from the fit. A good fit was defined
according to the match of the integrated intensities at all temperatures. b)
Fit result parameters of CsPbBr3 as function of temperature. The dashed lines
are guides to the eye.

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