Photo-induced enhancement of lattice fluctuations in metal-halide perovskites

The optoelectronic properties of metal-halide perovskites (MHPs) are affected by lattice fluctuations. Using ultrafast pump-probe spectroscopy, we demonstrate that in state-of-the-art mixed-cation MHPs ultrafast photo-induced bandgap narrowing occurs with a linear to super-linear dependence on the excited carrier density ranging from 1017 cm−3 to above 1018 cm−3. Time-domain terahertz spectroscopy reveals carrier localization increases with carrier density. Both observations, the anomalous dependence of the bandgap narrowing and the increased carrier localization can be rationalized by photo-induced lattice fluctuations. The magnitude of the photo-induced lattice fluctuations depends on the intrinsic instability of the MHP lattice. Our findings provide insight into ultrafast processes in MHPs following photoexcitation and thus help to develop a concise picture of the ultrafast photophysics of this important class of emerging semiconductors.

Comparison of deviation indicating that Eqn. S2a is more accurate than Eqn. S1 (black line). The deviation of Eqn. S2a depends on the value of npvk, however, it is not sensitive to it. For energies above Eopt, Eqn. S2a can be approximated as Eqn. S2b (red line).       Errors indicate the 95% confidence interval.

Supplementary Section 1: Ground-state absorption coefficient
Ground-state absorption measurements were performed using a PerkinElmer Lambda 950 UV/Vis/NIR spectrophotometer. To minimize the impact of reflection of light on the absorption spectra, the absorption coefficient is usually calculated by: where d is the sample thickness, R is the reflectance, and T is the transmittance. However, this equation is not sufficiently accurate for thin films, since it neglects internal reflections. Hence, we used a modified equation to extract the absorption spectra: where n2 is the averaged refractive index of perovskite films in the measured wavelength window. where 1, 2, and 3, respectively, refer to the air, the perovskite, and the substrate. r and t are the reflectance and transmittance, respectively. Specifically, 12 =̃1  ( 4 ) where R'3M and T'3M, respectively, represent the reflectance and transmittance of inverse propagation in the 3M system. Their value can be calculated by Eqn. S3 when swapping subscript 1 and 3. 31 = ( 3 − 1 3 + 1 ) 2 and 31 = 1 − 31 , respectively, represent the reflectance and transmittance at the interface of substrate and air.
The calculated R and T values when using Eqn. S4 are plotted in Supplementary Figure 1a.
The exact  and simulated  determined from Eqn. S1 and S2 are plotted in Supplementary   Figure 1b. The difference spectra (Supplementary Figure 1c) indicate that the error from Eqn. S2 is smaller than 2% above the optical bandgap (Eopt~1.6 eV). The validation of Eqn. S2 comes from the fact that n>>k and the dispersion of n is small in metal-halide perovskites. T3M can be simplified from Eqn. S3b when using n2>>k2 and a constant n2: 18 Eqn. S2a can be derived when setting b= T23T31 and c= R31/T31. A simpler expression (Eqn. S2b) can be obtained by neglecting the higher-order absorption term in Eqn. S7a (e -2d 0). This treatment increases the derivation below Eopt due to weak absorption (Supplementary Figure 1c).

Supplementary Section 2: 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 (D86~3 mm) 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 (D86~1 mm). The perovskite samples are stored in a nitrogen-filled chamber to protect from degradation, and photo-excited by 475 nm, 550 nm, and 675 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 (DR/R), then measured the transient absorption (DT/T).
Since Eqn. S2b derived from Eqn. S7a by neglecting the higher-order term provides a decent approximation of  above the optical bandgap, we start from Eqn. S2b to derive the relation between D and the measured DR/R, DT/T: Taking the total derivative and noticing that c is a constant and b is not sensitive to D: The second term at right side is small: 19 Because c is small (<0.05), the photo-induced change of the absorption coefficient can be expressed as: where R0 is the ground-state reflectivity spectrum. For large D, Eqn. S9 is no longer valid, hence D can be calculated by: The same expression of Eqn. S12 can be derived from Eqn. S1, indicating that internal reflections hardly affect the photo-induced change of  in perovskite thin films deposited on quartz substrates.
Since Eqn. S12 is equivalent to S11 for small D, all the D in this work were calculated by Eqn. S12. Supplementary Figure 4c is an example of the data processing.

Supplementary Section 3: Determination of the carrier density and thermal effect
The carrier densities generated by photoexcitation were calculated by: where pump is the absorption coefficient at pump photon energy (Eph). Fph is the photon flux calculated by deducting the fraction of surface-reflected photons from the total number of incident photons: where P is the pump power and Rpump is the reflectivity at Eph,. The factor 1500 denotes the pump laser pulse repetition rate in Hz. D represents the beam diameter of the pump beam measured by the beam profiler for the TA experiments (~3 mm, D86) and for the THz experiments (~4 mm, D86).

20
The heat accumulation effect can be ruled out in our experiments, as mentioned in the main text. The temperature rise caused by a single excitation pulse is smaller than 0.32 K for a carrier density of 3.510 18 cm -3 (i.e., the highest carrier density used in our TA experiments) when using 550 nm photons ( where DEex is the excess energy of the pump photons. N is carrier density. V is the volume in cm 3 . CMHP is the molar heat capacity of the MHP lattice, where we used the lower-bound value of 170 J/K/mol. Here, 410 21 is the number of unit cells in V=1 cm 3 . NA is the Avogadro constant.

Supplementary Section 4: Elliott fit to the absorption spectra
The ground-state absorption coefficient spectra were fitted by a convolution of broadening functions (Eqn. S18) and the following Elliott formula: Here, we evaluated three different broadening functions in the fit: Gaussian, hyperbolic-secant, and Voight (Gaussian convoluted with Lorentzian). Both the hyperbolic-secant distribution and Voight distribution result in excellent fits (Supplementary Figure 2). Since the hyperbolic-secant has only one parameter, we finally choose the hyperbolic-secant distribution as broadening function due to its simplicity.
The convolution of the continuum absorption and an arbitrary broadening function is: If x>>, then the broadening function is relatively narrow and we can consider (Rb0, t)t 1/2 as varying weakly. Thus, the integral can be avoided and we obtain: The deviation of this equation is smaller than 1% at E~1.8 eV for our FAMACs sample (top panel of Supplementary Figure 6). Therefore, the photo-induced change of the absorption coefficient can be calculated by: where DRb and DEbgr are the screened exciton binding energy and the photo-induced BGR due to many-body effects, respectively. Here a negative DEbgr represents bandgap narrowing. fe=1/[1+e (E-E F )/k B T e ] is the Fermi-Dirac distribution function accounting for the occupation probability of electrons in the conduction band, where EF is the quasi-Fermi level, kB is the Boltzmann constant, and Te is the absolute carrier temperature. Since the effective mass of holes is similar to that of electrons, we can assume that the occupation probability of holes in the valence band is symmetric to that of electrons in the conduction band.
When E-EF is much larger than kBTe, then it satisfies fe<<1, and Eqn. S21 can be further simplified. Applying a Taylor expansion and neglecting higher order terms we obtain: where 0 represents  (Rb0, x). Consequently, the relation between D and DEbgr can be approximated as:

Supplementary Section 5: Time-domain terahertz spectroscopy setup (td-THz)
Our td-THz 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 same TOPAS2 as used in the TA experiment. A motor equipped with a circular ND filter was used to change the pump fluence in the fluence dependent experiments.
The Hanning window was applied to the time-domain THz spectra to minimize the effect of frequency leakage caused by the discontinuity at boundaries. For terahertz waveforms with pulselike signals, the Hanning window is a good choice to resolve the frequency of interest by (1) removing boundary discontinuity in the time domain and (2) smoothing the broadband random noise in the frequency domain (Supplementary Figure 7).

Supplementary Section 6: Examination of the size effect in the THz spectra
Assuming that the terahertz electric field is Gaussian, the spot size of the terahertz probe can be calculated from the ratio of the maximum terahertz field transmitted through a 1-mm pinhole to its original terahertz field: The diameter (D86) of the THz probe is ~2.4 mm as calculated from the measured Tmax ~50%. We set the pump size to ~4 mm (D86, Supplementary Figure 8) to avoid any spot-size effects [Ref. S3] and to maximize the photon fluence. Supplementary Figure 9 shows that our THz experiment is free of spot size effects.

Supplementary Section 7: Optical phonon absorption in the THz spectra
The optical phonon effects on the photo-induced change of the photoconductivity spectra (D) are mainly from the infrared-active TO-phonons: a. Artifact due to data analysis: The equation above is derived based on the zeroth-order approximation [Ref. S4] that exhibits errors at frequencies with high dielectric background, yielding an artifact in the calculated D, especially at the peak of the TO-phonon resonance. Nevertheless, such artifact has a small