Ultrafast photo-induced enhancement of electron-phonon coupling in metal-halide perovskites


 In metal-halide perovskites (MHPs), the nature of organic cations affects both, the perovskite’s structure and its optoelectronic properties. Using ultrafast pump-probe spectroscopy, we demonstrate that in state-of-the-art mixed-cation MHPs ultrafast photo-induced bandgap narrowing occurs, and linearly depends on the excited carrier density in the range from 1016 cm− 3 to above 1018 cm− 3. Furthermore, time-domain terahertz (td-THz) photoconductivity measurements reveal that the majority of carriers are localized and that the localization increases with the carrier density. Both observations, the bandgap narrowing and carrier localization, can be rationalized by ultrafast (sub-2ps) photo-induced enhancement of electron-phonon coupling, originating from dynamic disorder, as clearly evidenced by the presence of a Debye relaxation component in the terahertz photoconductivity spectra. The observation of photo-induced enhancement of electron-phonon coupling and dynamic disorder not only provides specific insight into the polaron-strain distribution of excited states in MHPs, but also adds to the development of a concise picture of the ultrafast physics of this important class of semiconductors.

terahertz photoconductivity spectra. The observation of photo-induced enhancement of electronphonon coupling and dynamic disorder not only provides specific insight into the polaron-strain distribution of excited states in MHPs, but also adds to the development of a concise picture of the ultrafast physics of this important class of semiconductors.

Introduction
Understanding the excited carrier (photo) physics of metal-halide perovskite semiconductors is essential to unleash their optimum performance in optoelectronic applications, for instance, in light-emitting diodes and solar cells. In metal-halide perovskites Wannier-type excitons are the primary photo-generated states, since the small reduced electron mass (~0.1-0.15 me) 1-2 and high optical relative permittivity (εopt~4-6.5) 3 yield a Bohr radius larger than the perovskite's lattice constant. Based on the Wannier-exciton approximation, the exciton binding energy (Rb) can be estimated to ~54 meV when using εopt = 5 in the hydrogen model 2 . However, Rb~5-15 meV was determined based on temperature-dependent measurements of the absorption spectra of the classic CH3NH3PbI3 (MAPbI) perovskite, analyzed in the framework of Elliot theory [4][5][6] . Such small Rb can be explained when considering strong electron-phonon coupling, resulting from the perovskites' polar ionic crystal structure, which screens efficiently the electron-hole mutual attraction potential [7][8] . In fact, excitons are barely observed in metal halide perovskites at room temperature (where the thermal energy kBT is ~25 meV) due to efficient thermally-assisted dissociation. Consequently, free carriers are generated by photoexcitation, indicated by the quadratic carrier-density dependence of spontaneous photon emission 9 .
Nevertheless, the excited carriers are not entirely free in perovskite semiconductor films. First, the carrier's effective mobility is reduced by static disorder, caused by imperfections and thin-film inhomogeneities 10,11 . Second, the perovskite's ionic crystal lattice is soft. As a result, lattice distortions follow the motion of carriers, leading to the formation of quasiparticles known as polarons 12 . Furthermore, the energetic barrier for rotation of the organic cations present in the perovskite lattice is as small as ~20 meV 13 . Consequently, the organic cations can rotate around their lattice positions at room temperature, thereby increasing dynamic disorder 14 . Therefore, carriers can be scattered and trapped in localized states induced by static disorder, lattice distortions, and/or dynamic disorder 15 . The intimate relation between carrier localization and dynamic disorder has been revealed by density functional theory (DFT) [15][16][17][18][19][20] . It has been shown that dynamic disorder contributes to the discrepancy between the theoretically-predicted and experimentally-observed temperature dependence of polaron mobility [14][15] . The latest interpretation is that carrier localization is induced by dynamic disorder, but stabilized by lattice distortion 20 , or in other words, dynamic disorder is linked to polaron formation [18][19] , as also observed experimentally [21][22] . Since dynamic disorder scales with temperature, it enhances electron-phonon coupling and increases carrier localization at higher temperatures 23 , and thereby affects the polaron formation in the perovskite bulk 20 . However, the precise effect of dynamic disorder on electron-phonon coupling of excited states has remained elusive.
In this work, we employed transient absorption (TA) pump-probe spectroscopy to study electron-phonon coupling in an archetypical triple-cation perovskite film, namely FA0 .81 MA0 .14 Cs0 .05 PbI2 .55 Br0 .45 (FAMACs). This perovskite composition delivers a power conversion efficiency (PCE) higher than 20% in conjunction with excellent photostability in perovskite solar cells (PSCs) [24][25] . Carrier density-dependent TA spectra were measured at room temperature across carrier density ranges comparable to those observed at 1-sun illumination and above, precisely in the range from N~3.6×10 16 cm -3 to N~5.8×10 18 cm -3 . We developed a model to analyze the high-energy tail of the TA spectra and revealed photo-induced bandgap renormalization (BGR). Our results demonstrate that BGR concludes in ~2 ps and, unexpectedly, exhibits a linear dependence on the photo-excited carrier density. This linear dependence cannot be rationalized by carrier-carrier interactions, suggesting that electron-phonon coupling is its origin 26 . More precisely, for polaron formation, this anomalous BGR can only be rationalized by considering photo-enhanced electron-phonon coupling. We evidenced the presence of photoenhanced electron-phonon coupling by time-domain terahertz (tdTHz) spectroscopy. Specifically, we find that the fluence-dependent THz photoconductivity spectra can be described by a combination of the classical Drude model and the Debye relaxation model. The Debye relaxation component in the THz photoconductivity increases with carrier density, indicating an increase of the static-dielectric response, and thus enhancement of electron-phonon coupling at high carrier densities. The stabilization of this enhancement appears to be ultrafast, because the THz photoconductivity reaches its maximum in 2 ps. Since Debye relaxation is characteristic for dynamic disorder 27 , we ascribe the origin of the photo-enhanced electron-phonon coupling to dynamic disorder. The fraction of Drude-type carriers compared to the total number of photogenerated carriers decreases from ~10% to 5% as the carrier density increases from N~1×10 17 cm -3 to N~5×10 18 cm -3 , indicating that most carriers remain localized. This localization is even enhanced at high carrier densities, a consequence of increased dynamic disorder 15,20 . Due to the low fraction of Drude-type carriers, the effective carrier mobility is about an order smaller than the expected Drude-type carrier mobility, indicating that one has to account for fraction of localized carriers, when determining effective carrier mobilities. Finally, yet importantly, we relate the effective carrier density to the electron-phonon coupling constant and extract the photoenhanced fraction, which is in excellent agreement with the value predicted by linear BGR.
Our findings imply that dynamic disorder plays a major role in the physics of excited states in perovskites. More precisely, at low-fluence, dynamic disorder increases electron-phonon coupling and carrier localization on ultrafast timescales (<2 ps), providing polaron strain gradients that can drive halide segregation 28 . For carrier densities above a critical value, NC~5.3×10 18 cm -3 , the polaron strain gradients are gradually released due to spatial overlap of neighboring polarons, an effect that can be exploited to cure light-induced halide segregation 28 . Since the effective carrier mobility decreases with carrier density, the carrier diffusion length decreases at high carrier densities. Consequently, the tradeoff between these two effects has to be considered when designing high performance perovskite optoelectronic devices.
Transient absorption spectra -Analysis of the high-energy tail of the photobleach FAMAC films (thickness d~300 nm) were prepared by solution-processing of FAMAC solutions on spectroscopic-grade quartz substrates (see SI). Figure 1a shows the steady-state ground-state absorption spectrum, analyzed in the framework of the Elliott model, assuming the absorption (coefficient) consists of a linear combination of excitonic absorption (αX) and absorption of continuum states (αC) as reported earlier 4-6, 8, 29 . According to the Elliott theory, αC is enhanced by a factor ξ due to Coulomb attraction between electrons and holes 6 : where Rb0 is the exciton binding energy, and x=E-Eg, where Eg is the band gap. ξ approaches unity when Rb0 → 0, which implies the Coulomb enhancement of αC does not exist any longer once excitons are fully screened. Using ξ, the Elliott formula can be expressed by 6 : where A is a fitting parameter related to the transfer matrix elements 29 , δ is the Dirac delta function, √ is the normalized density of states in the conduction band assuming a parabolic shape (valid below ~1.82 eV in our sample). The first term accounts for αX, while the second term accounts for ξαC. To describe the room-temperature absorption spectra, a hyperbolic-secant broadening function accounting for thermal and inhomogeneous broadening was convoluted with Eqn. 2 (see SI), yielding Rb0 ~ 7.4 meV and Eg ~ 1.67 eV (Fig. 1a). The decomposed αX and ξαC are plotted in Figure 1a alongside αC. Clearly, the high-energy part of the absorption spectrum is very sensitive to ξ, the Coulomb screening.  against the carrier density. The red solid line is a linear fit to Ef as a function of log(N). Inset: timedependent Ef for a carrier density of ~4.3 ×10 18 cm -3 .
Next, we focus on the high-energy part of the TA spectra to explore the photo-induced BGR and underlying many-body effects (Fig. 1b). When analyzing the high-energy part of the spectra, we can safely neglect photo-induced broadening. For x larger than the width of the broadening function, the high-energy tails can be described by (see SI): where Rb and ∆Ebgr are the screened exciton binding energy and the photo-induced BGR due to many-body effects, respectively. Here a positive ∆Ebgr represents bandgap narrowing. fe=1/[1+e (E- is the Fermi-Dirac distribution function accounting for the occupation probability of electrons in the conduction band (inset in Fig. 1b), where Ef is the quasi-Fermi level, kB is the Boltzmann constant, and Te is the absolute electron temperature. Since the effective mass of holes is similar to that of electrons 30 , we assume that the occupation probability of holes in the valence band is symmetric to that of electrons in the conduction band. Compared to fits of the entire TA bleach region 31-32 , Eqn. 3 only uses four fitting parameters, namely Rb, ∆Ebgr, Ef and Te, owing to the intrinsic independence of the broadening function. Using the fitting results of the ground-state absorption spectrum (see Fig. S1, A=59451, Rb0=7.4 meV and Eg =1.67 eV), the high-energy tails (E~1.72 eV-1.82 eV) can be described very well by Eqn. 3 (Fig.1b). Here, Rb=Rb0 was used in the fitting, since even a small reduction in Rb resulted in poor fits.
The time-dependent Ef for N~4.3×10 18 cm -3 exhibits a fast picosecond rise followed by a slow rise in the range of tens of picoseconds (inset in Fig. 1c). This is related to hot-carrier cooling limited by the phonon bottleneck. If the value of Ef is smaller than Eg, then the electron density can be approximated by 6,31 : where m * is the effective electron mass and the other parameters have been defined above or have their usual meaning. Ef mainly depends on Te at early times when N is constant, thus its timedependent evolution mimics the effect of hot-carrier cooling. The reason for the cooling bottleneck has been summarized recently, 33 and it is not the focus of this work. Eqn. 4 also shows that Ef is linear in log(N) once hot-carrier cooling has concluded (Fig. 1c).  The green area represents the correlation (pulse width ~176 fs) of pump and probe laser pulses. Right axis: time evolution of electron temperature (Te) after photoexcitation. Te reaches its maximum soon after the instrument response (~352 fs), yet the terahertz kinetics indicate a maximum at ~2 ps. The rise of the terahertz kinetics observed after hot-carrier cooling has concluded is an evidence of polaron formation (shaded area).

Dependence of BGR on photo-excited carrier density
For N~4.3×10 18 cm -3 , ∆Ebgr assumes a plateau between ~2-5 ps after reaching the maximum (Fig. 2a), indicating photo-induced BGR is concluded in ~2 ps. The subsequent decrease of ∆Ebgr after ~5 ps is caused by the onset of carrier recombination. The stabilized ∆Ebgr is ~29 meV, similar to the value extracted from MAPbI films 31 . Since our data can only be fitted with Rb=Rb0, the nonzero ∆Ebgr implies that the excitonic level is red-shifted by ∆Ebgr (schematic in Fig. 2a). Such phenomenon cannot be explained by carrier-carrier interactions, which should leave the excitonic level virtually unchanged, because the photo-induced BGR and screening occur at the same time 34 . We note that the formation of bi-excitons may cause a large redshift of the excitonic level 35 .
However, this is not observed here. Hence, we conclude that ∆Ebgr must be caused by electronphonon interactions 26 .
It is noteworthy that Rb=Rb0 does not imply the absence of photo-induced screening, rather the photo-induced screening occurs on longer time scales, which do not affect Rb 14 . This time scale can be estimated by tscr>Rb0/h, where h is the Planck constant, yielding tscr>554 fs. Despite tscr larger than the time resolution of our TA spectroscopy setup (~176 fs), stabilized excitons are hardly detected due to very efficient thermal dissociation. In the framework of the hydrogen model, the effective exciton dielectric response (εX) at the ground state can be evaluated by: Rb0 =13.6mr/(meεX 2 ) eV, where 13.6 is the Rydberg constant and mr is the reduced electron mass. Here we obtain εX ~14.85 when using mr=0.12 me (ref. [1][2]. According to the Mott transition criterion of aB/λD=1.19, 36 where aB=0.0592meεX/mr is the effective Bohr radius of the exciton and λD= [εXET/(8πe 2 N)] 1/2 is the Debye screening length, using the thermal energy ET ~25 meV at room temperature, the estimated Mott density in our sample is NMott~2×10 16 cm -3 (Fig. 2b), indicating, hot excitons are separated instantaneously by phonon scattering if N>2×10 16 cm -3 .
The stabilized value of ∆Ebgr plotted versus the carrier density is shown in Figure 2b; likewise, the dependence cannot be explained by carrier-carrier interactions. In fact, in a doped semiconductor where carrier-carrier interactions are dominant, BGR should follow a power law dependence on the carrier density according to: ∆Ebgr ∝N k , where k~1/3, when electron-defect scattering is not significant [37][38] . However, Figure 2b shows k~1 in our sample, which cannot be explained by carrier-carrier interactions. The same results are found in other mixed-cation perovskites (see Fig. S2). The k=1 dependence can also result from electron-phonon interactions.
Perovskites are polar materials and their static-dielectric responses (εs) are much larger than the optical ones (εopt) 3,14 , resulting in pronounced Fröhlich electron-phonon coupling according to:  Figure 2c shows that the terahertz photoconductivity has reached only half of its maximum when hot-carrier cooling has concluded, indicating electron-phonon interactions are the main reason for the rise in the first 1-2 ps. This phenomenon has been ascribed unambiguously to polaron formation in single-cation perovskites 41 .
Generally, Epol does not contribute to ∆Ebgr, because it is contained in the ground-state absorption spectrum from which Eg is extracted (Fig. 1a). However, if aep is enhanced by photoexcitation, Epol can contribute to ∆Ebgr according to: Δ~2Δ Ω This enhancement can be enabled by the presence of coupling between photon-induced dynamic disorder and the lattice distortion [21][22] .

Photo-enhanced dielectric response
The photo-enhanced dielectric response was first revealed by electrical experiments 42-43 , however, the conclusions have remained debated 44 . Here we provide further evidence based on tdTHz spectroscopy. Following the measurement of the tdTHz spectrum without prior photoexcitation (ETHz), the photo-induced tdTHz spectra (∆ETHz) were recorded at 5 ps after photoexcitation (inset in Fig. 3a). The differential tdTHz spectra in the frequency domain (∆ETHz/ETHz) can be obtained by Fast Fourier Transformation (FFT), and subsequently the photoinduced change of the dielectric response (∆εTHz) can be calculated by 45 : where ε0 is the vacuum dielectric constant, c the speed of light, nsub~2.13 the refractive index of the quartz substrate in the terahertz region, d is the sample thickness, ∆σ the photo-induced change of the photoconductivity, and ω is the angular frequency.
where εs is the static dielectric constant and τr is the Debye relaxation time. τr is related to the phonon activation energy (Ea) and the thermal energy by τr=τ0*exp(Ea/ET) with constant τ0. Since ET is determined by the lattice temperature, which does not change significantly, τr can be considered constant in our experiments. Eqn. 8 implies that dynamic disorder contributes to ∆εTHz.
More precisely, the photo-induced low energy polarizations give rise to the dielectric response.
Next, we turn to the complex photoconductivity change ∆σ, which is shown frequently in THz works ( Fig. 3b-3c). At low carrier densities, ∆σ 1 (the real part of ∆σ) is virtually frequency independent (Fig 3b) and ∆σ 2 (the imaginary part of ∆σ) is close to zero (Fig. 3c). These spectral signatures indicate, the photoconductivity is mediated by free charges that undergo high rate scattering events 46 . At high carrier densities, ∆σ 1 drops at the low frequency side and ∆σ 2 shows a zero-crossing. Such behavior has been explained in the framework of the Drude-Smith model (DS model) which involves carrier localization 10 : where e is the unit charge of electron, ND is the density of Drude carriers, τsc is the carrier's momentum scattering time. cj is a parameter (-1≤cj≤0) that accounts for photoconductivity renormalization caused by the j-th scattering event. Here, the scattering time for each scattering event is considered unchanged. cj=-1 implies full back scattering (no conduction), while cj=0 implies no backscattering (Drude conduction). A negative cj shifts the Drude response to high frequency.
The DS model fits well at low carrier densities, however, it cannot describe ∆σ 2 at high carrier densities (Fig. 3c), indicating localized carriers exhibit different scattering times due to dynamic disorder. Since ∆σ 2 is directly related to the real part of ∆εTHz (Eqn. 7), we combined the Drude model and Debye relaxation model (DD model) to describe our data: where ∆εs is the photo-induced change of the static dielectric response originated from dynamic disorder. The first term on the right side of the equation accounts for the Drude photoconductivity, the second term accounts for the photoconductivity resulting from dynamic disorder. ∆σ 1 and ∆σ 2 can be well fitted globally by the DD model (Fig. 3b-3c), with τsc~16.7 fs and τr ~343 fs. The shaded area in Figure 3b represents the photoconductivity originating from dynamic disorder, which tends to become zero at ω=0, indicating dynamic disorder does not support long range photoconductivity.

Discussion
The fraction of Drude carriers among all photo-generated carriers (ND/N) is shown in Figure 4a (top panel), alongside the ratio of ∆εs/N (bottom panel in Fig. 4a). We find that ND/N is only ~5%-10% depending on the carrier density, similar to the value determined for MAPbI films 10 and MAPbI single crystals 47 . ∆εs/N is almost constant, indicating a linear increase of ∆εs as carrier density increases. However, the uncertainty is significant at low carrier densities, because of the poor signal-to-noise ratio. To confirm that ND/N and ∆εs/N do not change significantly at low carrier densities, the effective carrier mobility (µeff) was calculated by µeff =µND/N (Fig. 4b), where µ is the mobility of Drude carriers: µ=eτsc/m *~1 27.8 cm 2 /V/s. For comparison, the effective mobility was also directly calculated from the frequency-averaged ∆σ 1 by µ σ =σavg/Ne (Fig. 4b).
We expected that µ σ >>µeff if ND/N → 1 at low carrier densities, however, Figure 4b shows µ σ is only slightly larger than µeff for all carrier densities investigated, indicating that there is no significant change of ND/N and ∆εs/N at carrier densities around N~10 17 cm -3 . The low value of ND/N indicates µeff mainly originates from a small fraction of photo-generated carriers, consistent with the complicated localization nature in MHPs 15 . Our finding also suggests that µ σ is a good approximation of µeff without the need to extract it from fits of the terahertz photoconductivity.    48 or if the exciton binding energy is high (~200 meV) 49 , stabilized excitons are unlikely to be present in our experiments (Fig. 2b). The enhanced carrier localization coincides with the increase of the shaded area in Figure 3b, confirming the intimate relation between carrier localization and dynamic disorder. Since a positive ∆εs will result in an increase of aep (Eqn. 5), in the weak-coupling regime, µeff can be related to ∆aep by using the concept of polaron mobility 50 : where µ0 and aep0 are the polaron mobility and Fröhlich electron-phonon coupling constant at N→0, respectively, and the term 1+aep0/6 indicates the polaron mass is enhanced by a factor of ~aep/6 compared to the effective electron mass in the rigid lattice 50 . Combining Eqn. 5 and Eqn. 11, only three unknown parameters remain: ΩLO, εopt and εs. For simplicity, we fit µeff by using ∆εs/N=2.6×10 -18 , ΩLO/h=3.7 THz, and εopt~4-6.5 (Fig. 4b). The fitted εs is ~9-12.6, much smaller than the theoretically predicted value 3 . The discrepancy can arise from the uncertainty of ΩLO and because electron-phonon coupling is primarily mediated by optical phonons 17,51 , which only contribute weakly to the static dielectric response 3 .
The coupling time is ultrafast (<2 ps, see rise in Fig. 2c) and related to the Debye relaxation time τr ~343 fs. Since ∆aep is only determined by the carrier-density dependence of the dynamic disorder, its value is marginal at carrier densities below N~10 17 cm -3 (Fig. 4c), indicating ∆aep can be neglected for MHP solar cells that work under 1-sun illumination. However, ∆aep exacerbates the polaron strain gradients and thereby the halide segregation 28 . Under high irradiance, ∆aep reduces the carrier mobility and thus the carrier diffusion length.
Next, we use the volume ratio of polarons to evaluate the polaron strain gradient (gpol). We note that, when two adjacent polarons merge, the polaron strain cancels in the overlapping part: where rpol is the Feynman polaron radius, 52 and dpol is the distance between two neighboring polarons. The two can be approximated by: where v=3+2aep/9 and w=3 are variable parameters in units of ΩLO. 52 The calculated rpol and dpol are given in SFig. 3, and gpol is plotted in Fig. 4d. When the carrier density increases, gpol reaches its maximum at a critical carrier density of NC~5.3×10 18 cm -3 . When N>NC, gpol drops due to the spatial overlap of adjacent polarons. Therefore, high irradiance can be used to cure light-induced halide segregation and to alter the perovskite bandgap 28 .
In conclusion, triple-cation-mixed MHP films (FAMACs) were investigated by TA spectroscopy and tdTHz spectroscopy. We developed a model to analyze the high-energy tails of the TA spectra, taking into account the exciton binding energy and the photo-induced BGR. We demonstrate that the photo-induced BGR reveals ultrafast (<2 ps) photo-enhanced electron-phonon coupling, which has its origin in dynamic disorder, as evidenced by the Debye relaxation component observed in the terahertz photoconductivity spectra. Furthermore, we determined that photo-generated carriers are highly localized. Importantly, the effective carrier mobility can be approximated from the frequency-averaged terahertz photoconductivity measurements without further fits. Finally, we extracted the photo-enhanced electron-phonon coupling constant as a function of the carrier density, and we revealed ultrafast coupling (<2 ps) between dynamic disorder and the lattice distortion. Our findings provide insights into the ultrafast photophysics of perovskites, specifically the polaron strain distribution of excited states in perovskite devices and as a function of the photogenerated carrier density.

AUTHOR INFORMATION
The authors declare no competing financial interests.    r pol for ε opt =6.5 Figure S3: Polaron radius (rpol) and the distance between two neighboring polars (dpol) at different carrier densities, calculated using Eqn.13 in the main manuscript.

Section 1. Materials and sample preparation
Materials: PbI2 and PbBr2 were purchased from TCI. FAI and MABr were purchased from Dyesol. CsI, RbI and all anhydrous solvents (DMF, DMSO, chlorobenzene) were purchased from Sigma-Aldrich. SnO2 colloid precursor was obtained from Alfa Aesar, the particles were diluted by H2O and isopropanol to 2.67 %. All chemicals were used without further purification.

Section 2. Ground-state absorption coefficient
Ground-state absorption measurements were performed using a PerkinElmer Lambda 950 UV/Vis/NIR spectrophotometer. To reduce the impact of reflection on the absorption spectra, the absorption coefficient was calculated by: where d is the sample thickness, R is the reflectivity, and T is the transmissivity.

Section 3. Transient absorption (TA) spectroscopy
Our TA setup uses a commercial Ti:sapphire amplifier operating at 800 nm with a repetition rate of 3 kHz as laser source. Its pulse width (FWHM) is compressed to ~125 fs. Two optical parametric amplifiers (OPA) are used to tune the laser wavelength. The white-light probe is generated by 1300 nm laser (from TOPAS1) with a CaF2 crystal that mounted on a continuously moving stage, which enables us to generate a super-continuum pulses with a spectral range from 350 to 1100 nm. The pump laser (from TOPAS2) is chopped to 1.5 KHz and delayed by an automated mechanical delay stage (Newport linear stage IMS600CCHA) from -400 ps to 8 ns. Pump and probe beams were overlapped on the front surface of the sample, and their spot sizes were measured by a beam viewer (Coherent, LaserCam-HR II) to make sure the pump beam was about three times larger than the probe beam. The perovskite samples are stored in a nitrogenfilled chamber to protect from degradation, and photo-excited by 475 nm in this work. The probe beam was guided to a custom-made prism spectrograph (Entwicklungsbüro Stresing) where it was dispersed by a prism onto a 512 pixel complementary metal-oxide semiconductor (CMOS) linear image sensor (Hamamatsu G11608-512DA). In order to account for the reflection, we first measured the transient reflection (R and ∆R/R), then measured the transient absorption (∆T/T) and calculated the photo-induced change of absorption coefficient by:

Section 4. Time-resolved terahertz spectroscopy setup (TRTS)
Our TRTS setup uses the same Ti:sapphire amplifier as the TA setup. The THz emitter and detector are two 1 mm thick <110> oriented zinc telluride (ZnTe) crystals. All the THz related optics were placed in a closed chamber, which was continuously purged with pure nitrogen gas. Perovskite samples were excited by 550 nm laser pulses obtained from the TOPAS2 in the TA setup. A rotation motor mounted with a circular ND filter was used to change the pump fluence in the fluence dependent experiments.

Section 5. Determination of the carrier density
The carrier densities injected by photoexcitation were calculated by: where F is the photon flux calculated by deducting the surface reflected photons from the total incident photons: where P is the pump power. The number 1500 denotes the number of pulses that can pass the optical chopper in one second. DFWHM~3.2 mm represents the full width at half maximum (FWHM) of the pump beam measured by the beam profiler.
The FWHM of the terahertz probe is calculated from the ratio of the maximum terahertz intensity transmitted through a 1-mm pinhole to its original intensity: