# Infrared optical absorption of Fr$\ddot{o}$hlich polaron in metal halide
perovskites

Yu Cui  Tianjin Key Laboratory of Low Dimensional Materials Physics and
Preparing Technology, Department of Applied Physics, School of Science,
Tianjin University, Tianjin 300354 China  Xiao-Yi Liu  Tianjin Key Laboratory
of Low Dimensional Materials Physics and Preparing Technology, Department of
Applied Physics, School of Science, Tianjin University, Tianjin 300354 China
Xu-Fei Ma  Tianjin Key Laboratory of Low Dimensional Materials Physics and
Preparing Technology, Department of Applied Physics, School of Science,
Tianjin University, Tianjin 300354 China  Jia-Pei Deng  Tianjin Key Laboratory
of Low Dimensional Materials Physics and Preparing Technology, Department of
Applied Physics, School of Science, Tianjin University, Tianjin 300354 China
Yi-Yan Liu  Tianjin Key Laboratory of Low Dimensional Materials Physics and
Preparing Technology, Department of Applied Physics, School of Science,
Tianjin University, Tianjin 300354 China  Zhi-Qing Li  Tianjin Key Laboratory
of Low Dimensional Materials Physics and Preparing Technology, Department of
Applied Physics, School of Science, Tianjin University, Tianjin 300354 China
Zi-Wu Wang<EMAIL_ADDRESS>Tianjin Key Laboratory of Low Dimensional
Materials Physics and Preparing Technology, Department of Applied Physics,
School of Science, Tianjin University, Tianjin 300354 China

###### Abstract

The formation of Fr$\ddot{o}$hlich polaron in metal halide perovskites,
arising from the charge carrier-longitudinal optical (LO) phonon coupling, has
been proposed to explain their exceptional properties, but the effective
identification of polaron in these materials is still a challenge task.
Herein, we theoretically present the infrared optical absorption of
Fr$\ddot{o}$hlich polaron based on Huang-Rhys model. We find that multiphonon
overtones are appeared as the energy of incident photon matches the multiple
LO phonons, wherein the average phonon numbers of a polaron can be directly
evaluated by the order of the strongest overtone. These multiphonon structures
sensitively depend on the scale of electronic distribution in the ground state
and the dimensionality of the perovskite materials, which gives the
enlightenment for the effective modulation of competing processes between the
polaron formation and carrier cooling. Moreover, the order of the strongest
overtone shifts to the higher ones with temperature, providing a potential
proof of the carriers mobility affected by LO phonons scattering. The present
model not only suggests a direct way to verify Fr$\ddot{o}$hlich polaron, but
also enriches the understanding of the polaron properties in metal halide
perovskites.

metal halide perovskites, Fr$\ddot{o}$hlich polaron, Huang-Rhys factor,
infrared optical absorption

##  I Introduction

Fr$\ddot{o}$hlich polaron (or large polaron) formation is originated from the
coupling between a macroscopic electronic field induced by the polar optical
phonon modes in solid and a charge carrier. So the polaron was also called as
the special carrier dressed by the $``phonon$ $cloud"$ (the induced nuclear
polarization), reflecting by the average phonon numbersA. S. Mishchenko
(2000); J. T. Devreese (2007, 2009). This type of polarons has aroused more
and more enthusiasm in metal halide perovskites (MHP) because it has been used
to unlock the key puzzles regarding the exceptional propertiesD. Ghosh (2020);
K. Miyata (2017); A. Mahata (2019); Y. Lan (2019), such as the moderate
carrier mobility, large carrier diffusion length, long carrier lifetime and
high defect tolerance. Therefore, the identification on this polaron in MHP
becomes one of the crucial problems in both experimental and theoretical
studies.

In the past few years, the formation of Fr$\ddot{o}$hlich polaron in
perovskite materials has been probed in spectral measurements: following the
polaron formation dynamics in time-resolved optical Kerr effect spectroscopyK.
Miyata (2017), ultrafast Terahertz (THz) spectroscopyE. Cinquanta (2019) as
well as time-resolved two-photon photoemission and transient reflectance
spectroscopiesT. J. S. Evans (2018); examining Pb-I structural dynamics in
time-domain Raman spectroscopyM. Park (2017); studying the electronic
structure in angle-resolved photoelectron spectroscopyM. Puppin (2020). On the
other hand, large polarons can be also speculated by other measurements such
as the mobility of charge carrier scaling with temperature in the transportM.
J. Schilcher (2021) and Hall effect measurementsY. Chen (2016). In the
theoretical aspect, large polarons forming predominantly from the distortion
or deformation of the crystal structure with the first-principle calculation
have also been widely studiedD. Meggiolaro (2020); A. J. Neukirch (2016);
M.-A. Husanu (2020). Despite these experimental and theoretical evidences have
confirmed the formation of large polaron in perovskite materials, the detail
knowledge and direct distinction for the “phonon cloud” of a polaron, which is
the crucial ingredient determining the nature of the polaron, have not been
well understood to date.

Figure 1:  The schematic diagram of optical absorption of the polaron, where
$\hbar\omega$ is the incident photon energy and $\hbar{\omega_{LO}}$ is the
longitudinal optical (LO) phonon energy.

In this paper, we theoretically study the infrared optical absorption of
Fr$\ddot{o}$hlich polaron in the three-dimensional (3D) and two-dimensional
(2D) MHP materials stemming from the strong electron-longitudinal optical (LO)
phonon modes coupling. We find that the “phonon cloud” of polaron can be
directly reflected by multiple LO phonons structure of the optical absorption
spectroscopy, in which the average phonon numbers described very well by the
dimensionless Huang-Rhys factors are directly linked to the order of the
strongest absorption peak. Moreover, the strongest absorption peak shifts to
the higher order with decreasing the electronic distribution radius in the
ground state, indicating that the range of the electronic distribution plays
an important role in determining the electron-LO phonon coupling. Meanwhile,
the temperature dependence of the polaron absorption processes is also
discussed.

##  II The theoretical model

Figure 2:  Huang-Rhys factors in CH3NH3PbI3 (a) and CH3NH3PbBr3 (b) as a
function of the electronic distribution radius in the ground state for
different dimensionalities. $\mathcal{S}^{2D}$ and $\mathcal{S}^{3D}$
represent Huang-Rhys factors in two- and three-dimensional perovskite
materials, respectively.

As schemed in Fig. 1, the absorption of light by polarons can be described as
follows: when the incident photon energy $\hbar\omega$ matches the multiple LO
phonons energy $p\hbar\omega_{LO}$, the polaron system will undergo
transitions from the ground state
$|{\Psi_{i}}\rangle=|{\psi_{1s}}\rangle|{\chi_{i{n^{\prime}_{\bf{q}}}}}\rangle$
with energy $E_{i}$ towards a series of excited states
$|{\Psi_{f}}\rangle=|{\psi_{1s}}\rangle|{\chi_{f{n_{\bf{q}}}}}\rangle$ with
energy $E_{f}=E_{i}+p\hbar\omega_{LO}$. $|\psi_{1s}\rangle$ is the ground
state of polaron, and $|{\chi_{i{n^{\prime}_{\bf{q}}}}}\rangle$
$(|{\chi_{f{n_{\bf{q}}}}}\rangle)$ is the initial (final) lattice wave
function, where $n^{\prime}_{\bf{q}}$ ($n_{\bf{q}}$) is the occupation number
for phonons with the wave vector $\bf{q}$. Note that the optical absorption
discussed here is one polaron behavior rather than many-polaron gas. As is
typical of multiphonon transition process, it was excellently studied by the
Huang-Rhys modelK. Huang (1950, 1981); Z.-W. Wang (2019). By means of this
well-known model, the probability of optical absorption for the light with
frequency $\omega$ can be expressed asE. Kartheuser (1980); M. Matsuura
(1980); Z. Cheng (1999)

$\displaystyle\mathcal{F}(\hbar\omega)$ $\displaystyle=$
$\displaystyle\sum\limits_{f}{\frac{{4{\pi^{2}}{e^{2}}\hbar\omega}}{{3{m^{2}}{\hbar^{2}}\varepsilon
c}}}{\left|{\left\langle{{\psi_{1s}}}\right|{\bf{r}}\left|{{\psi_{1s}}}\right\rangle}\right|^{2}}$
(1)
$\displaystyle\times{\left|{\left\langle{{{\chi_{fn_{\bf{q}}}}}}\mathrel{\left|{\vphantom{{{\chi_{fn}}}{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}}\right.\kern-1.2pt}{{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}\right\rangle}\right|^{2}}\delta({E_{f}}-{E_{i}}-\hbar\omega),$

$e$ is the charge carrier, $\hbar\omega$ is the photon energy, $m$ is the
effective mass of the polaron, $\varepsilon$ is the permittivity, $c$ is the
velocity of light, and $\bf{r}$ is the position variables for polaron.

Figure 3: Multiple LO phonons infrared optical absorption spectra in 3D
CH3NH3PbI3 (a), 3D CH3NH3PbBr3 (b), 2D CH3NH3PbI3 (c) and 2D CH3NH3PbBr3 (d)
for different electronic distribution radiuses in the ground state, at $T$ =
77 K.

For the ground state of polaron, a Gaussian wave function is consideredJ. T.
Devreese (1999):

$\psi_{1s}^{\lambda}(r)={\mathcal{C}_{\lambda}}\exp(-\frac{{{r^{2}}}}{{4a_{0}^{2}}}),$
(2)

where the normalization constant
${\mathcal{C}_{\lambda}}={[1/(\sqrt{2\pi}{a_{0}})]^{3/2}}$ for $\lambda$ =
$3{\rm D}$ and ${\mathcal{C}_{\lambda}}={1/(\sqrt{2\pi}{a_{0}})}$ for
$\lambda$ = $2{\rm D}$ ($\lambda$ = $3{\rm D}$ and $\lambda$ = $2{\rm D}$
represent the 3D and 2D materials, respectively.). $a_{0}$ represents the
radius of electronic distribution in the ground state. Upon substitution of
Eq. (2), Eq. (1) will be simplified as

$\displaystyle{\mathcal{F}^{\lambda}}(\hbar\omega)$ $\displaystyle=$
$\displaystyle\sum\limits_{f}{{\Re_{\lambda}}}a_{0}^{2}\hbar\omega{\left|{\left\langle{{{\chi_{fn_{\bf{q}}}}}}\mathrel{\left|{\vphantom{{{\chi_{fn}}}{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}}\right.\kern-1.2pt}{{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}\right\rangle}\right|^{2}}$
(3) $\displaystyle\times\delta({E_{f}}-{E_{i}}-\hbar\omega).$

${\Re_{3D}}=32\pi{e^{2}}/(3{m^{2}}{\hbar^{2}}\varepsilon c)$ and
${\Re_{2D}}=2\pi^{3}{e^{2}}/(3{m^{2}}{\hbar^{2}}\varepsilon c)$ are the
optical constants. The overlap between the initial
$|{\chi_{i{n^{\prime}_{\bf{q}}}}}\rangle$ and final
$|{\chi_{f{n_{\bf{q}}}}}\rangle$ lattice vibration produced by the lattice
displacement occurring during the transition process is responsible for the
multiple phonons structure, which is linked to the electron-phonon coupling.
Substituting the expression of the square of the overlap integral
${\left|{\left\langle{{{\chi_{fn_{\bf{q}}}}}}\mathrel{\left|{\vphantom{{{\chi_{fn}}}{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}}\right.\kern-1.2pt}{{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}\right\rangle}\right|^{2}}$
shown in the Appendix A into Eq. (3), we finally convert the optical
absorption spectra of polaron involving the participation of $p$ phonons at
finite temperature into

$\displaystyle{\mathcal{F}^{\lambda}}(\hbar\omega)$ $\displaystyle=$
$\displaystyle{\Re_{\lambda}}a_{0}^{2}\hbar\omega{(\frac{{\overline{n}_{T}+1}}{{\overline{n}_{T}}})^{p/2}}\exp[-{\mathcal{S}^{\lambda}}(2\overline{n}_{T}+1)]$
$\displaystyle\times{I_{p}}(2{\mathcal{S}^{\lambda}}\sqrt{\overline{n}_{T}(\overline{n}_{T}+1)})\exp[-\frac{{{{(\hbar\omega-p\hbar{\omega_{LO}})}^{2}}}}{{2{\sigma^{2}}}}].$

LO phonon is mainly taken into account because it couples with the charge
carriers in Fr$\ddot{o}$hlich mechanism as the main source of lattice
distortion for MHP materials in not only theories but also experimentsC.
Quarti (2014); C. M. Iaru (2021, 2017, 2021).
$\overline{n}_{T}=1/({e^{\hbar{\omega_{LO}}/{k_{B}}T}}-1)$ is the phonon
number occupation function where $k_{B}$ and $\hbar{\omega_{LO}}$ are the
Boltzmann constant and LO phonon energy, respectively; $I_{p}$ is the $p$th
order imaginary argument Bessel function with the phonon number $p$; $\sigma$
represents the line-width of the phonon overtone. The strength of electron-
phonon coupling has been characterized using the Huang-Rhys (HR) factor
${\mathcal{S}^{\lambda}}$, which plays a nontrival role in determining the
multiphonon processes. In the Fr$\ddot{o}$hlich mechanismJ. T. Devreese
(1999); D. Emin (1999); Y. Cui (2021), HR factor can be calculated by

${\mathcal{S}^{\lambda}}=\sum\limits_{\bf{q}}\ell_{\lambda}(q){\left({\frac{\hbar}{{2{m^{*}}{\omega_{LO}}}}}\right)^{1/2}}\int{\psi_{1s}^{{\lambda}*}(r){e^{i{\bf{q}}\cdot{\bf{r}}}}}\psi_{1s}^{\lambda}(r)d{\bf{r}}$
(5)

$\ell_{3D}(q)=4\pi\alpha/(V{q^{2}})$ and $\ell_{2D}(q)=2\pi\alpha/(A{q})$. The
detailed derivations for HR factors, charge carrier-LO phonon coupling, the
corresponding defination and expression of parameters are shown in the
Appendix B. In the numerical calculation processes, we choose typical MHP
CH3NH3PbI3 (MAPbI3) and CH3NH3PbBr3 (MAPbBr3) as examples to discuss the
optical absorption processes of the polaron. Throughout this paper, the line-
width $\sigma$ = 3 meV is assumed. The adopted values of other parameters for
two types of materials are shown in Table I.

Figure 4: The temperature dependences of multiple LO phonons structure for the optical absorption spectra in 3D MAPbI3 (a) and 3D MAPbBr3 (b), respectively, at $a_{0}=0.4R_{0}$.  Table 1:  The adopted parameters for the theoretical calculations. $m_{0}=9.1\times{10^{-31}}$ kg is the free electron mass. Parameters were taken from Refs. A. D. Wright (2016) and M. Sendner (2016).  Parameter | MAPbI3 | MAPbBr3
---|---|---
LO phonon energy ($\hbar\omega_{LO}$) | 11.5 meVA. D. Wright (2016) | 15.3 meVA. D. Wright (2016)
Coupling constant ($\alpha$) | 1.72M. Sendner (2016) | 1.69M. Sendner (2016)
Effective mass ($m^{*}$) | 0.104$m_{0}$M. Sendner (2016) | 0.117$m_{0}$M. Sendner (2016)
Polaron radius ($R_{0}$) | 51 ÅM. Sendner (2016) | 43 ÅM. Sendner (2016)

##  III Results and discussion

HR factor was originally proposed by Huang and Rhys for F-centres in ionic
crystals in 1950K. Huang (1950), which denotes the average phonon numbers
around F-centres, reflecting the magnitude of lattice distortion arising from
the strong coupling between electronic states and lattice vibration (phonons).
Herein, HR factor is invoked to describe the “phonon cloud” of the polaron
with relaxation energy ${\mathcal{S}^{\lambda}}\hbar\omega_{LO}$. The
dependences of HR factors on the electronic distribution radius in the ground
state in different dimensionalities are shown in Figs. 2(a) and (b) for MAPbI3
and MAPbBr3, respectively. One can see that HR factors are enlarged from 3D to
2D, which are stemming from that both the reduced dielectric screening and
increased degree of quantum confinement are held responsible for the
enhancement of electron-phonon coupling. On the other hand, HR factor exhibits
a negative correlation with $a_{0}$. But HR factors vary smoothly as $a_{0}$
exceeds 0.6$R_{0}$ ($R_{0}$ is the polaron radius). This indicates that
outside this region the localization feature of electrons has little effect on
the electron-phonon coupling. Amounts of researchers have paid attention to
the calculation of the formation energy of the large polaron. The obtained
values were 12 meV for 3D MAPbI3 and 16$\pm$2 meV for 3D MAPbBr3 in a tight-
binding model fitted from the first-principle calculation and temperature-
dependent absorption spectroscopyF. Zheng (2019, 2019), respectively. These
numerical values are in good agreement with the relaxation energy
${\mathcal{S}^{3D}}\hbar\omega_{LO}\approx 13.8$ meV for 3D MAPbI3 and
${\mathcal{S}^{3D}}\hbar\omega_{LO}\approx 18.2$ meV for 3D MAPbBr3 at
$a_{0}\geq 0.6R_{0}$, implying that the formation energy can be well described
by the relaxation energy. As for 2D perovskite species, though there is little
direct information about polarons in the experiments, the discoveries of the
2D exciton polarons and self-trapped excitons can reveal the significant
polaronic characterJ. Yin (2017); A. Kandada (2020); S. Li (2019); J. Li
(2019). The numerical values of HR factors for 2D perovskites in Fig. 2 are
expected to provide insight for the identification on these quasi-particles.
In addition, it is shown that HR factors in MAPbBr3 are slightly smaller than
that in MAPbI3. It is ascribed to the essence that the distortion of the Pb-Br
bond is harder than that of the Pb-I bond determined by the bond length, which
is reflected by the coupling parameter $\alpha$ as listed in Table I. We have
noticed that the effective phonon modes and relaxation energy could be
obtained from the first-principles calculationsL. Shi (2012, 2015); Y. Xiao
(2020), gaining the more accurate evaluations for HR factors. Our results will
also provide an important theoretical reference for these calculations.

On the basis of these theoretical values of HR factors, the optical absorption
spectra of polaron for MAPbI3 and MAPbBr3 are plotted in Fig. 3 for different
electronic distribution radiuses in the ground state. We first give the
spectra for MAPbI3 a detailed discussion. In Fig. 3(a), the energy resonance
absorption will occur when the energy of the incident photon matches the
multiple LO phonons, showing the multiphonon overtones in the infrared (long
wavelength) region of spectroscopy, where the strongest overtone corresponds
to the most probable state of polaron. And the intensities of overtones follow
an asymmetric Gaussian distribution, which are in agreement with the previous
measurements in experimentsK. H. Kim (1998); E. Cinquanta (2019). It is worth
noting that the order of the strongest overtone $p_{max}$ satisfies the
condition of $p_{max}\approx{\mathcal{S}^{3D}}+0.5$, such as
$\mathcal{S}^{3D}\approx 2.42$ corresponding to $p_{max}=3$ at
$a_{0}=0.4R_{0}$, but the condition shifts to
$p_{max}\approx{\mathcal{S}^{2D}}+1$ for the 2D perovskites, such as
$\mathcal{S}^{2D}\approx 3.81$ corresponding to $p_{max}=5$ at
$a_{0}=0.4R_{0}$, shown in Fig. 3(c). This means that the average phonon
number of polaron could be estimated directly from the absorption spectra. In
terms of this significant outcome, it can be used to explain the overtones
along with a linear decay from the first to the higher orders in the
multiphonon Roman scattering and THz time-dominant spectroscopy for CsPbBr3
($\mathcal{S}^{3D}=0.39$)C. M. Iaru (2021). The pronounced redshift behaviors
for the position of the spectral weight with the increase of the electron
localization are shown, specifically manifested in Fig. 3(a) as (1) the number
of the observable overtones reduces from seven at ${a_{0}}=0.4{R_{0}}$, to
five at ${a_{0}}=0.8{R_{0}}$; (2) the most probable state transits from the
third order at $a_{0}=0.4R_{0}$ to the second order at $a_{0}=0.8R_{0}$. These
phenomena are the direct reflections of the variation of the electron-phonon
coupling (HR factors) linked to the charge carrier localization, showing the
polaron to be more stable or to be more easily taken off phonon dressing. The
similar trends for MAPbBr3 are illustrated in Figs. 3(b) and (d). Recently,
several experiments have proved that the large polaron formation drastically
suppresses the hot carriers coolingS. A. Bretschneider (2018); T. J. S. Evans
(2018); P. P. Joshi (2019); G. Kaur (2019). For instance, an envelope of
Cs4PbBr6 lattice shell was provided over the CsPbBr3 nanocrystalsG. Kaur
(2019). The enhanced strength of the electron-phonon coupling in such shell
material facilitates the formation of polarons with large relaxation energy,
and thus depresses the carriers cooling processes. It is, therefore, concluded
that if more phonon overtones are appeared in the absorption spectra, the
polaron effect will be more pronounced, which could be served as a criterion
that how to judge these perovskites materials being suitable for the potential
applications in photovoltaic (perfer to the strong polaron effect) or
photoelectronic (demanding the fast carrier cooling) devices.

The temperature dependences of optical absorption in the 3D MAPbI3 and 3D
MAPbBr3 are shown in Figs. 4(a) and 4(b), respectively. These multiphonon
overtones will be divided into two regions: the low orders region including
the overtones are in the range of $1\leq{p}\leq\mathcal{S}^{3D}+2$ and the
rest of overtones being categorized as the high orders region. In the former
region, the intensities of overtones decrease monotonically with temperature.
On the contrary, the latter region shows the opposite variational trends and
even much higher orders of overtones will be thermally activated. It can be
explained by that more thermally activated LO phonons become available with
temperature, resulting in (1) the polaron states with more phonons are prone
to reach the most probable states, and (2) the orders of overtones increase.
The blueshift of the most probable states leads to the shift of the whole
absorption spectra towards the higher energy, showing the downside of the
relative probability in the low order region and the upside trend for the high
order region as given in Figs. 4(a) and (b). More importantly, the polaron,
dressed by more thermally activated LO phonons, indicates that the polaron
effects are actively strengthened at higher temperature. In the perovskite
materials, the long carrier diffusion length does not imply a high charge
carrier mobility, yet a detailed understanding of its fundamental mechanisms
is still outstanding. According to our theoretical study, the polarons become
significantly “heavier” with temperature, which is not beneficial to
transport, especially at room temperature. Thus it provides a potential
explanation for the moderate carrier mobility in perovskites, which supports
the theoretical predictions about LO phonon mode scattering influencing
mobilityM. Sendner (2016); A. Thongnum (2021). Therefore, it may be more
appropriate to elucidate the effective mobility by the polaron model in
perovskite materialsM. Sendner (2016); F. Zheng (2019); G. Kaur (2019); J. M.
Frost (2017); C. Motta (2017). In turn, the measurements of the carrier
mobility in experiments could also track the trace of polaronsT. M. Brenner
(2017). It is additionally revealed that the overtones keep stable outline
even at room temperature, implying the polaron states have a very robust
lifetime, which will benefit not only the experimental observation but also
the potential exploitation of polaron properties in perovskite materials.
Finally, we must emphasize that these perovskite materials undergo two
structure phase transitions with temperature, which has an influence on the
strength of the electron-phonon coupling and phonon energyP. S. Whitfield
(2016); L. Liu (2019); M. Keshavarz (2019). These effects are not taken into
account in the present paper.

##  IV Conclusions

In summary, we theoretically study the infrared optical absorption
spectroscopy of Fr$\ddot{o}$hlich polaron in metal halide perovskites. We find
that (1) the multiphonon structure can be used as the fingerprint for the
polaron, where the average phonon number in the “phonon cloud” of the polaron
is reflected by the order of the strongest overtone in the absorption spectra;
(2) the multiphonon structure depends on both the dimensionality of materials
and the charge carrier localization sensitively, which give the enlightenment
for controlling the competing processes between the polaron formation and
carrier cooling; (3) the “phonon cloud” around the polaron will become larger
as the temperature increases and thus affect the carrier mobility, especially
at room temperature, which is consistent with the experimental measurements.

## ACKNOWLEDGMENT

This work was supported by National Natural Science Foundation of China (Grant
Nos. 11674241 and 12174283).

##  Appendix A The calculation of the overlap integrals

Within the adiabatic approximation, the square of the overlap integral of the
lattice wave function considering the electron-phonon coupling can be written
asE. Kartheuser (1980); M. Matsuura (1980); M. Soltani (1995)

$\displaystyle{\left|{\left\langle{{{\chi_{fn_{\bf{q}}}}}}\mathrel{\left|{\vphantom{{{\chi_{fn_{\bf{q}}}}}{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}}\right.\kern-1.2pt}{{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}\right\rangle}\right|^{2}}$
$\displaystyle=$
$\displaystyle{{{\left|\prod\limits_{\bf{q}}{\int{{\chi_{{n_{\bf{q}}}}}(Q)\exp[{\textsl{g}_{\bf{q}}}a_{\bf{q}}^{\dagger}-\textsl{g}_{\bf{q}}^{*}{a_{\bf{q}}}]{\chi_{{n^{\prime}_{\bf{q}}}}}(Q)dQ}}\right|}^{2}}},$

$Q$ describes the normal mode of lattice vibration,
${\textsl{g}_{\bf{q}}}={{{{\cal
M}_{q}}{\rho_{\bf{q}}}}\mathord{\left/{\vphantom{{{{\cal
M}_{q}}{\rho_{\bf{q}}}}{(\hbar{\omega_{LO}}}}}\right.\kern-1.2pt}{(\hbar{\omega_{LO}}}})$
is the variational parameter with the coupling element $\mathcal{M}_{q}$ and
the electronic charge density $\rho_{\bf{q}}$. $a_{\bf{q}}^{\dagger}$
($a_{\bf{q}}$) is the creation (annihilation) operator of a LO phonon with
wave vector $\bf{q}$, and ${\chi_{{n_{\bf{q}}}}}(Q)$
(${\chi_{{n^{\prime}_{\bf{q}}}}}(Q)$) is the free LO phonon states, where
$n^{\prime}_{\bf{q}}$ ($n_{\bf{q}}$) is the occupation number for phonons. The
space translation operator
$\exp[{{\textsl{g}_{\bf{q}}}a_{\bf{q}}^{\dagger}}-\textsl{g}_{\bf{q}}^{*}{a_{\bf{q}}}]$
reflects the variation of the equilibrium position on account of the electron-
phonon coupling.

The creation and annihilation operators can be expressed asD. Goguenheim
(1990)

$a_{\bf{q}}^{\dagger}=\sqrt{\frac{{{\omega_{LO}}}}{{2\hbar}}}(Q-i\frac{P}{{{\omega_{LO}}}}),$
(7)
${a_{\bf{q}}}=\sqrt{\frac{{{\omega_{LO}}}}{{2\hbar}}}(Q+i\frac{P}{{{\omega_{LO}}}}),$
(8)

where $P$ is the momentum operator. Upon substitution of Eqs. (A2) and (A3),
Eq. (A1) can be of the form

$\displaystyle{\left|{\left\langle{{{\chi_{fn_{\bf{q}}}}}}\mathrel{\left|{\vphantom{{{\chi_{fn_{\bf{q}}}}}{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}}\right.\kern-1.2pt}{{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}\right\rangle}\right|^{2}}$
$\displaystyle=$
$\displaystyle{\left|\prod\limits_{\bf{q}}{\int{{\chi_{{n_{\bf{q}}}}}(Q)[\sum\limits_{n=0}^{\infty}{\frac{{{\mathcal{A}^{n}}}}{{n!}}{Q^{n}}}{\chi_{n^{\prime}_{\bf{q}}}}(Q)]dQ}}\right|^{2}}$
$\displaystyle=$
$\displaystyle{\left|\prod\limits_{\bf{q}}{\int{{\chi_{{n_{\bf{q}}}}}[{\chi_{n^{\prime}_{\bf{q}}}}+\mathcal{A}Q{\chi_{n^{\prime}_{\bf{q}}}}+\frac{{{\mathcal{A}^{2}}}}{2}{Q^{2}}{\chi_{n^{\prime}_{\bf{q}}}}+\cdots]dQ}}\right|^{2}},$

where
$\mathcal{A}={[{\omega_{LO}}/(2\hbar)]^{1/2}}({\textsl{g}_{\bf{q}}}-\textsl{g}_{\bf{q}}^{*})$.
For the change in phonon number between $n_{\bf{q}}$ and
$n^{\prime}_{\bf{q}}$, only transitions in which the phonon number changes at
most one contribute to the transition probability. Therefore, only three cases
are taken into account for the change of the phonon number:

(1) for the case ${n_{\bf{q}}}={n^{\prime}_{\bf{q}}}+1$, one can get

$\displaystyle\int{{\chi_{{n^{\prime}_{\bf{q}}+1}}}[\mathcal{A}Q{\chi_{{n^{\prime}_{\bf{q}}}}}]dQ}$
$\displaystyle=$
$\displaystyle\frac{{\sqrt{{n^{\prime}_{\bf{q}}}+1}}}{2}{({\textsl{g}_{\bf{q}}}-\textsl{g}_{\bf{q}}^{*})};$

(2) for the case ${n_{\bf{q}}}={n^{\prime}_{\bf{q}}}-1$, one can get

$\displaystyle\int{{\chi_{{n^{\prime}_{\bf{q}}}-1}}[\mathcal{A}Q{\chi_{{n^{\prime}_{\bf{q}}}}}]dQ}$
$\displaystyle=$
$\displaystyle\frac{{\sqrt{{n^{\prime}_{\bf{q}}}}}}{2}{({\textsl{g}_{\bf{q}}}-\textsl{g}_{\bf{q}}^{*})};$

(3) for the case ${n_{\bf{q}}}={n^{\prime}_{\bf{q}}}$, one can get

$\displaystyle\int{{\chi_{{n^{\prime}_{\bf{q}}}}}{\chi_{{n^{\prime}_{\bf{q}}}}}dQ}+\frac{{{\mathcal{A}^{2}}}}{2}\int{{\chi_{{n^{\prime}_{\bf{q}}}}}{Q^{2}}{\chi_{{n^{\prime}_{\bf{q}}}}}dQ}+\cdots$
$\displaystyle=$ $\displaystyle
1+\frac{{2{n^{\prime}_{\bf{q}}}+1}}{8}{({\textsl{g}_{\bf{q}}}-\textsl{g}_{\bf{q}}^{*})^{2}}+\cdots.$

When the transition process involves $p$ phonons, the contribution of these
three cases should be made up of $k$ oscillators going down by one quantum and
$k+p$ oscillators going up by one quantum. Therefore, the square of the
overlap integral can be rewritten as

$\displaystyle{\left|{\left\langle{{{\chi_{fn_{\bf{q}}}}}}\mathrel{\left|{\vphantom{{{\chi_{fn_{\bf{q}}}}}{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}}\right.\kern-1.2pt}{{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}\right\rangle}\right|^{2}}$
$\displaystyle=$
$\displaystyle{\left|{\frac{{\sqrt{{n^{\prime}_{{{\bf{q}}_{1}}}}+1}}}{2}({\textsl{g}_{{{\bf{q}}_{1}}}}-\textsl{g}_{{{\bf{q}}_{1}}}^{*})}\right|^{2}}{\left|{\frac{{\sqrt{{n^{\prime}_{{{\bf{q}}_{2}}}}+1}}}{2}({\textsl{g}_{{{\bf{q}}_{2}}}}-\textsl{g}_{{{\bf{q}}_{2}}}^{*})}\right|^{2}}\cdots{\left|{\frac{{\sqrt{{n^{\prime}_{{{\bf{q}}_{p+k}}}}+1}}}{2}({\textsl{g}_{{{\bf{q}}_{p+k}}}}-\textsl{g}_{{{\bf{q}}_{p+k}}}^{*})}\right|^{2}}$
(10)
$\displaystyle\times{\left|{\frac{{\sqrt{{n^{\prime}_{{{\bf{q}}_{1}}}}}}}{2}({\textsl{g}_{{{\bf{q}}_{1}}}}-\textsl{g}_{{{\bf{q}}_{1}}}^{*})}\right|^{2}}{\left|{\frac{{\sqrt{{n^{\prime}_{{{\bf{q}}_{2}}}}}}}{2}({\textsl{g}_{{{\bf{q}}_{2}}}}-\textsl{g}_{{{\bf{q}}_{2}}}^{*})}\right|^{2}}\cdots{\left|{\frac{{\sqrt{{n^{\prime}_{{{\bf{q}}_{k}}}}}}}{2}({\textsl{g}_{{{\bf{q}}_{k}}}}-\textsl{g}_{{{\bf{q}}_{k}}}^{*})}\right|^{2}}$
$\displaystyle\times{\left|{\prod\limits_{\bf{q}}{[1+\frac{{2{n^{\prime}_{\bf{q}}}+1}}{8}{{({\textsl{g}_{\bf{q}}}-\textsl{g}_{\bf{q}}^{*})}^{2}}+\cdots]}}\right|^{2}}.$

For the given $k$ and $p$, there are a large number of distinct transitions,
corresponding to different choices of the oscillators. Their total
contribution can be obtained by summing Eq. (A5) over all the indices, and
dividing afterwards by $k!(k+p)!$. Then Eq. (A5) becomesK. Huang (1950, 1981);
Z.-W. Wang (2019)

$\displaystyle{\left|{\left\langle{{{\chi_{fn_{\bf{q}}}}}}\mathrel{\left|{\vphantom{{{\chi_{fn_{\bf{q}}}}}{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}}\right.\kern-1.2pt}{{{\chi_{i{n^{\prime}_{\bf{q}}}}}}}\right\rangle}\right|^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{{k!(k+p)!}}{\left\\{{\sum\limits_{\bf{q}}{({n^{\prime}_{\bf{q}}}+1)}{\textsl{g}_{\bf{q}}}\textsl{g}_{\bf{q}}^{*}}\right\\}^{p+k}}{\left\\{{\sum\limits_{\bf{q}}{{n^{\prime}_{\bf{q}}}}{\textsl{g}_{\bf{q}}}\textsl{g}_{\bf{q}}^{*}}\right\\}^{k}}\exp\left\\{{\sum\limits_{\bf{q}}{[\frac{{2{n^{\prime}_{\bf{q}}}+1}}{4}{{({\textsl{g}_{\bf{q}}}-\textsl{g}_{\bf{q}}^{*})}^{2}}]}}\right\\}$
(11) $\displaystyle=$
$\displaystyle\exp[-\mathcal{S}(2\overline{n}_{T}+1)]\sum\limits_{k}{\frac{{{{[\mathcal{S}(\overline{n}_{T}+1)]}^{p+k}}{{[\mathcal{S}\overline{n}_{T}]}^{k}}}}{{k!(k+p)!}}}$
$\displaystyle=$
$\displaystyle{(\frac{{\overline{n}_{T}+1}}{{\overline{n}_{T}}})^{p/2}}\exp[-\mathcal{S}(2\overline{n}_{T}+1)]{I_{p}}(2\mathcal{S}\sqrt{\overline{n}_{T}(\overline{n}_{T}+1)}).$

$\overline{n}_{T}=1/({e^{\hbar{\omega_{LO}}/{k_{B}}T}}-1)$ is the phonon
number occupation function where $k_{B}$ and $\hbar{\omega_{LO}}$ are the
Boltzmann constant and LO phonon energy, respectively; $I_{p}$ is the
imaginary argument Bessel function with the phonon number $p$; ${\cal
S}=\sum\limits_{\bf{q}}{{{{{\left|{\cal
M}_{q}\right|}^{2}}{{\left|{{\rho_{\bf{q}}}}\right|}^{2}}}\mathord{\left/{\vphantom{{{{\left|{{M_{q}}}\right|}^{2}}{{\left|{{\rho_{\bf{q}}}}\right|}^{2}}}{{{(\hbar{\omega_{LO}})}^{2}}}}}\right.\kern-1.2pt}{{{(\hbar{\omega_{LO}})}^{2}}}}}={\left|{{\textsl{g}_{\bf{q}}}}\right|^{2}}$
is the Huang-Rhys factor.

##  Appendix B The calculation of the Huang-Rhys factor

The electron-LO phonon coupling in the Fr$\ddot{o}$hlich mechanism can be
expressed asJ. T. Devreese (1999); D. Emin (1999)

$\displaystyle{\cal H}_{e-p}^{\lambda}=\sum\limits_{\bf{q}}{({\cal
M}_{q}^{\lambda*}a_{\bf{q}}^{\dagger}{e^{-i{\bf{q}}\cdot{\bf{r}}}}+{\cal
M}_{q}^{\lambda}{a_{\bf{q}}}{e^{i{\bf{q}}\cdot{\bf{r}}}})}$ (12)

with

${{\left|{{\mathcal{M}^{\lambda}_{q}}}\right|^{2}}}=\left\\{\begin{array}[]{rcl}&{\frac{{{{(\hbar{\omega_{LO}})}^{2}}}}{{{q^{2}}}}\frac{{4\pi\alpha}}{V}{(\frac{\hbar}{{2{m^{*}}{\omega_{LO}}}})^{1/2}}},&(\lambda=3D)\\\
&{\frac{{{{(\hbar{\omega_{LO}})}^{2}}}}{q}\frac{{2\pi\alpha}}{A}{(\frac{\hbar}{{2{m^{*}}{\omega_{LO}}}})^{1/2}}},&(\lambda=2D)\\\
\end{array}\right..$ (13)

$V$ is the crystal volume, $A$ is the crystal area, and $\alpha$ denotes the
Fr$\ddot{o}$hlich coupling strength. The source of the lattice polarization is
the charge distribution of the polaron described by the Fourier transform

$\rho_{\bf{q}}^{\lambda}{\rm{=}}\int{{{\left|{\psi_{1s}^{\lambda}(r)}\right|}^{2}}}{e^{i{\bf{q}}\cdot{\bf{r}}}}d{\bf{r}},(\lambda=3D,2D).\\\
$ (14)

Substituting Eq. (B2) as well as (B3) into the expression of Huang-Rhys factor
${\mathcal{S}}$, the Huang-Rhys factor in the Fr$\ddot{o}$hlich mechanism can
be calculated byM. Soltani (1995); K. Huang (1950)

$\displaystyle{\mathcal{S}^{3D}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\bf{q}}{\frac{{{{\left|{{\cal
M}_{q}^{3D}}\right|}^{2}}{{\left|{\rho_{\bf{q}}^{3D}}\right|}^{2}}}}{{{{\left({\hbar{\omega_{LO}}}\right)}^{2}}}}}$
(15) $\displaystyle=$
$\displaystyle\sum\limits_{\bf{q}}\frac{{4\pi\alpha}}{{V{q^{2}}}}{\left({\frac{\hbar}{{2{m^{*}}{\omega_{LO}}}}}\right)^{1/2}}\int{\psi_{1s}^{{3D}*}(r){e^{i{\bf{q}}\cdot{\bf{r}}}}}\psi_{1s}^{3D}(r)d{\bf{r}}$
$\displaystyle=$ $\displaystyle\frac{{\alpha{R_{0}}}}{{\sqrt{\pi}{a_{0}}}}$

for the 3D materials and

$\displaystyle{\mathcal{S}^{2D}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\bf{q}}{\frac{{{{\left|{{\cal
M}_{q}^{2D}}\right|}^{2}}{{\left|{\rho_{\bf{q}}^{2D}}\right|}^{2}}}}{{{{\left({\hbar{\omega_{LO}}}\right)}^{2}}}}}$
(16) $\displaystyle=$
$\displaystyle\sum\limits_{\bf{q}}\frac{{2\pi\alpha}}{{A{q}}}{\left({\frac{\hbar}{{2{m^{*}}{\omega_{LO}}}}}\right)^{1/2}}\int{\psi_{1s}^{{2D}*}(r){e^{i{\bf{q}}\cdot{\bf{r}}}}}\psi_{1s}^{2D}(r)d{\bf{r}}$
$\displaystyle=$ $\displaystyle\frac{{\sqrt{\pi}\alpha{R_{0}}}}{{2{a_{0}}}}$

for the 2D materials. $a_{0}$ is the electronic distribution radius in the
ground state, and ${R_{0}}={[\hbar/(2{m^{*}}{\omega_{LO}})]^{1/2}}$ is the
polaron radius.

## References

  * A. S. Mishchenko (2000) A. S. Mishchenko, N. V. Prokof’ev, A. Sakamoto, and B. V. Svistunov, Diagrammatic quantum Monte Carlo study of the Fröhlich polaron, Phys. Rev. B 62, 6317 (2000).
  * J. T. Devreese (2007) J. T. Devreese, Fröhlich polarons from 0D to 3D: Concepts and recent developments, J. Phys.: Condens. Matter 19, 255201 (2007).
  * J. T. Devreese (2009) J. T. Devreese and A. S. Alexandrov, Fröhlich polaron and bipolaron: Recent developments, Rep. Prog. Phys. 72, 066501 (2009).
  * K. Miyata (2017) K. Miyata, T. L. Atallah, and X.-Y. Zhu, Lead halide perovskites: Crystal-liquid duality, phonon glass electron crystals, and large polaron formation, Sci. Adv. 3, e1701469 (2017).
  * Y. Lan (2019) Y. Lan, B. J. Dringoli, D. A. Valverde-Ch$\acute{a}$vez, C. S. Ponseca Jr., M. Sutton, Y. He, M. G. Kanatzidis, and D. G. Cooke, Ultrafast correlated charge and lattice motion in a hybrid metal halide perovskite, Sci. Adv. 5, eaaw5558 (2019).
  * A. Mahata (2019) A. Mahata, D. Meggiolaro, and F. D. Angelis, From Large to Small Polarons in Lead, Tin, and Mixed Lead-Tin Halide Perovskites, J. Phys. Chem. Lett. 10, 1790 (2019).
  * D. Ghosh (2020) D. Ghosh, E. Welch, A. J. Neukirch, A. Zakhidov, and, S. Tretiak, Polarons in Halide Perovskite: A Perspective, J. Phys. Chem. Lett. 11, 3271 (2020).
  * K. Miyata (2017) K. Miyata, D. Meggiolaro, M. T. Trinh, P. P. Joshi, E. Mosconi, S. C. Jones, F. D. Angelis, and X.-Y. Zhu, Large polarons in lead halide perovskites, Sci. Adv. 3, e1701217 (2017).
  * E. Cinquanta (2019) E. Cinquanta, D. Meggiolaro, S. G. Motti, M. Gandini, M. J. P. Alcocer, Q. A. Akkerman, C. Vozzi, L. Manna, F. D. Angelis, A. Petrozza, and S. Stagira, Ultrafast THz Probe of Photoinduced Polarons in Lead-Halide Perovskites, Phys. Rev. Lett. 122, 166601 (2019).
  * T. J. S. Evans (2018) T. J. S. Evans, K. Miyata, P. P. Joshi, S. Maehrlein, F. Liu, and X.-Y. Zhu, Competition Between Hot-Electron Cooling and Large Polaron Screening in CsPbBr3 Perovskite Single Crystals, J. Phys. Chem. C 122, 13724 (2018).
  * M. Park (2017) M. Park, A. J. Neukirch, S. E. Reyes-Lillo, M. Lai, S. R. Ellis, D. Dietze, J. B. Neaton, P. Yang, S. Tretiak, and R. A. Mathies, Excited-state vibrational dynamics toward the polaron in methylammonium lead iodide perovskite, Nat. Commun. 9, 2525 (2018).
  * M. Puppin (2020) M. Puppin, S. Polishchuk, N. Colonna, A. Crepaldi, D. N. Dirin, O. Nazarenko, R. De Gennaro, G. Gatti, S. Roth, T. Barillot, L. Poletto, R. P. Xian, L. Rettig, M. Wolf, R. Ernstorfer, M. V. Kovalenko, N. Marzari, M. Grioni, and M. Chergui, Evidence of Large Polarons in Photoemission Band Mapping of the Perovskite Semiconductor CsPbBr3, Phys. Rev. Lett. 124, 206402 (2020).
  * M. J. Schilcher (2021) M. J. Schilcher, P. J. Robinson, D. J. Abramovitch, L. Z. Tan, A. M. Rappe, D. R. Reichman, and D. A. Egger, The Significance of Polarons and Dynamic Disorder in Halide Perovskites, ACS Energy Lett. 6, 2162 (2021).
  * Y. Chen (2016) Y. Chen, H. T. Yi, X. Wu, R. Haroldson, Y. N. Gartstein, Y. I. Rodionov, K. S. Tikhonov, A. Zakhidov, X.-Y. Zhu, and V. Podzorov, Extended carrier lifetimes and diffusion in hybrid perovskites revealed by Hall effect and photoconductivity measurements, Nat. Commun. 7, 12253 (2016).
  * A. J. Neukirch (2016) A. J. Neukirch, W. Nie, J. Blancon, K. Appavoo, H. Tsai, M. Y. Sfeir, C. Katan, L. Pedesseau, J. Even, J. J. Crochet, G. Gupta, A. D. Mohite, and S. Tretiak, Polaron Stabilization by Cooperative Lattice Distortion and Cation Rotations in Hybrid Perovskite Materials, Nano Lett. 16, 3809 (2016).
  * D. Meggiolaro (2020) D. Meggiolaro, F. Ambrosio, E. Mosconi, A. Mahata, and F. D. Angelis, Polarons in Metal Halide Perovskites, Adv. Energy Mater. 10, 1902748 (2019).
  * M.-A. Husanu (2020) M.-A. Husanu, L. Vistoli, C. Verdi, A. Sander, V. Garcia, J. Rault, F. Bisti, L. L. Lev, T. Schmitt, F. Giustino, A. S. Mishchenko, M. Bibes, and V. N. Strocov, Electron-phonon dichotomy of charge carriers in perovskite oxides, Commun. Phys. 3, 62 (2020).
  * K. Huang (1950) K. Huang and A. Rhys, Theory of light absorption and non-radaitive transitions in F-centres, Proc. R. Soc. A 204, 406 (1950).
  * K. Huang (1981) K. Huang, Lattice relaxation and multiphonon transitions, Contemp. Phys 22, 599 (1981).
  * Z.-W. Wang (2019) Z.-W. Wang, Y. Xiao, J.-P. Deng, Y. Cui, and Z.-Q. Li, Multiphonon Raman scattering mediated by the exciton states in monolayer transition metal chalcogenides, Phys. Rev. B 100, 125308 (2019).
  * E. Kartheuser (1980) E. Kartheuser and R. Evrard, Radiative recombination of donor-acceptor pairs in polar semiconductors, Phys. Rev. B 21, 648 (1980).
  * M. Matsuura (1980) M. Matsuura and H. Büttner, Optical properties of excitons in polar semiconductors: Energies, oscillator strengths, and phonon side bands, Phys. Rev. B 21, 679 (1980).
  * Z. Cheng (1999) Z. Cheng, Unified quantum field theory of light absorption by defect centers, Phys. Rev. B 60, 15747 (1999).
  * J. T. Devreese (1999) J. T. Devreese, Fröhlich Polarons-Lecture Course Including Detailed Theoretical Derivations, Universiteit Antwerpen, Belgium, 2016, e-print arXiv: 1012.4576v6.
  * C. Quarti (2014) C. Quarti, G. Grancini, E. Mosconi, P. Bruno, J. M. Ball, M. M. Lee, H. J. Snaith, A. Petrozza, and F. D. Angelis, The Roman Spectrum of the CH3NH3PbI3 Hybrid Perovskite: Interplay of Theory and Experiment, J. Phys. Chem. Lett. 5, 279 (2014).
  * C. M. Iaru (2017) F. Brivio, J. M. Frost, J. M. Skelton, A. J. Jackson, O. J. Weber, M. T. Weller, A. R. Goni, A. M. A. Leguy, P. R. F. Barnes, and A. Walsh, Lattice Dynamics and Vibrational Spectra of the Orthorhombic, Tetragonal, and Cubic Phases of Methylammonium Lead Iodide, Phys. Rev. B 92, 144308 (2015).
  * C. M. Iaru (2021) C. M. Iaru, J. J. Geuchies, P. M. Koenraad, D. Vanmaekelbergh, and A. Y. Silov, Strong carrier-phonon coupling in lead halide perovskite nanocrystals, ACS Nano 11, 11024 (2017).
  * C. M. Iaru (2021) C. M. Iaru, A. Brodu, N. J. J. van Hoof, S. E. T. ter Huurne, J. Buhot, F. Montanarella, S. Buhbut, P. C. M. Christianen, D. Vanmaekelbergh, C. de M. Donega, J. G. Rivas, P. M. Koenraad, and A. Y. Silov, Fr$\ddot{o}$hlich interaction dominated by a single phonon mode in CsPbBr3, Nat. Commu. 12, 5844 (2021).
  * D. Emin (1999) D. Emin, Polarons, Cambridge University Press (1999).
  * Y. Cui (2021) Y. Cui, X.-Y. Liu, S.-Y. Ji, Y. Sun, J.-P. Deng, X.-F. Ma, Z.-Q. Li, and Z.-W. Wang, Energy resonance transfer between quantum defects in mental halide perovskites, J. Phys. Chem. Lett. 12, 11182 (2021).
  * A. D. Wright (2016) A. D. Wright, C. Verdi, R. L. Milot, G. E. Eperon, M. A. Pérez-Osorio, H. J. Snaith, F. Giustino, M. B. Johnston, and L. M. Herz, Electron-phonon coupling in hybrid lead halide perovskites, Nat. Commun. 7, 11755 (2016).
  * M. Sendner (2016) M. Sendner, P. K. Nayak, D. A. Egger, S. Beck, C. Müller, B. Epding, W. Kowalsky, L. Kronik, H. J. Snaith, A. Pucci, and R. Lovrin$\breve{c}$i$\acute{c}$, Optical phonons in methylammonium lead halide perovskites and implications for charge transport, Mater. Horiz. 3, 613 (2016).
  * F. Zheng (2019) C. Wolf, J.-S. Kim, and T.-W. Lee, Structural and thermal disorder of solution-processed CH3NH3PbBr3 hybrid perovskite thin films, ACS Appl. Mater. Interfaces 9, 10344 (2017).
  * F. Zheng (2019) F. Zheng and L.-W. Wang, Large polaron formation and its effect on electron transport in hybrid perovskites, Energy Environ. Sci. 12, 1219 (2019).
  * J. Yin (2017) J. Yin, H. Li, D. Cortecchia, C. Soci, and J. Brédas, Excitonic and Polaronic Properties of 2D Hybrid Organic-Inorganic Perovskites, ACS Energy Lett. 2, 417 (2017).
  * S. Li (2019) S. Li, J. Luo, J. Liu, and J. Tang, Self-Trapped Excitons in All-Inorganic Halide Perovskites: Fundamentals, Status, and Potential Applications, J. Phys. Chem. Lett. 10, 1999 (2019).
  * J. Li (2019) J. Li, J. Wang, J. Ma, H. Shen, L. Li, X. Duan, and D. Li, Self-trapped state enabled filterless narrowband photodetections 2D layered perovskite single crystals, Nat. Commun. 10, 806 (2019).
  * A. Kandada (2020) A. R. S. Kandada and C. Silva, Exciton Polarons in Two-Dimensional Hybrid Metal-Halide Perovskites, J. Phys. Chem. Lett. 11, 3173 (2020).
  * L. Shi (2012) L. Shi and L.-W. Wang, $Ab$ $initio$ calculations of deep-level carrier nonradiative recombination rates in bulk semiconductors, Phys. Rev. Lett. 109, 245501 (2012).
  * L. Shi (2015) L. Shi, K. Xu, and L.-W. Wang, Comparative study of $ab$ $initio$ nonradiative recombination rate calculations under different formalisms, Phys. Rev. B 91, 205315 (2015).
  * Y. Xiao (2020) Y. Xiao, Z.-W. Wang, L. Shi, X.-W. Jiang, S.-S. Li, and L.-W. Wang, Anharmonic multi-phonon nonradiative transition: an $ab$ $initio$ calculation approach, Sci. China Phys. Mech. Astron. 63, 277312 (2020).
  * K. H. Kim (1998) K. H. Kim, J. H. Jung, and T. W. Noh, Polaron absorption in a perovskite manganite La0.7Ca0.3MnO3, Phys. Rev. Lett. 81, 1517 (1998).
  * S. A. Bretschneider (2018) S. A. Bretschneider, I. Ivanov, H. I. Wang, K. Miyata, X. Zhu, and M. Bonn, Quantifying Polaron Formation and Charge Carrier Cooling in Lead-Iodide Perovskites, Adv. Mater. 30, 1707312 (2018).
  * P. P. Joshi (2019) P. P. Joshi, S. F. Maehrlein, and X. Zhu, Dynamic Screening and Slow Cooling of Hot Carriers in Lead Halide Perovskites, Adv. Mater. 31, 1803054 (2019).
  * G. Kaur (2019) G. Kaur, K. J. Babu, N. Ghorai, T. Goswami, S. Maiti, and H. N. Ghosh, Polaron-Mediated Slow Carrier Cooling in a Type-1 3D/0D CsPbBr3@CsPbBr6 Core-Shell Perovskite System, J. Phys. Chem. Lett. 10, 5302 (2019).
  * A. Thongnum (2021) A. Thongnum, R. Pingaew, and U. Pinsook, Impact of the polar optical phonon and alloy scattering on the charge-carrier mobilities of FA0.83Cs0.17Pb(I1-xBrx)3 hybrid perovskites, Phys. Chem. Chem. Phys. 23, 27320 (2021).
  * J. M. Frost (2017) J. M. Frost, Calculating polaron mobility in halide perovskites, Phys. Rev. B 96, 195202 (2017).
  * C. Motta (2017) C. Motta and S. Sanvito, Electron-Phonon Coupling and Polaron Mobility in Hybrid Perovskites from First Principles, J. Phys. Chem. C 122, 1361 (2018).
  * G. Kaur (2019) A. Thongum and U. Pinsook, Polaron transport in hybrid CH3NH3PbI3 perovskite thin films, Nanoscale 12, 14112 (2020).
  * T. M. Brenner (2017) T. M. Brenner, D. A. Egger, A. M. Rappe, L. Kronik, G. Hodes, and D. Cahen, Are Mobilities in Hybrid Organic-Inorganic Halide Perovskites Actually “High”?, J. Phys. Chem. Lett. 6, 4754 (2015).
  * P. S. Whitfield (2016) P. S. Whitfield, N. Herron, W. E. Guise, K. Page, Y. Q. Cheng, I. Milas, and M. K. Crawford, Structures, Phase Transitions and Tricritical Behavior of the Hybrid Perovskite Methyl Ammonium Lead Iodide, Sci. Rep. 6, 35685 (2016).
  * L. Liu (2019) L. Liu, R. Zhao, C. Xiao, F. Zhang, F. Pevere, K. Shi, H. Huang, H. Zhong, and I. Sychugov, Size-Dependent Phase Transition in Perovskite Nanocrystals, J. Phys. Chem. Lett. 10, 5451 (2019).
  * M. Keshavarz (2019) M. Keshavarz, M. Ottesen, S. Wiedmann, M. Wharmby, R. Küchler, H. Yuan, E. Debroye, J. A. Steele, J. Martens, N. E. Hussey, M. Bremholm, M. B. J. Roeffaers, and J. Hofkens, Tracking Structural Phase Transitions in Lead-Halide Perovskites by Means of Thermal Expansion, Adv. Mater. 31, 1900521 (2019).
  * M. Soltani (1995) M. Soltani, M. Certier, R. Evarad, and E. Kartheuser, Photoluminescence of CdTe doped with arsenic and antimony acceptors, J. Appl. Phys. 78, 5626 (1995).
  * D. Goguenheim (1990) D. Goguenheim and M. Lannoo, Theoretical and experimental aspects of the thermal dependence of electron capture coefficients, J. Appl. Phys. 68, 1059 (1990).