## Tracking carrier and exciton dynamics in mixed-cation lead mixed-halide perovskite thin films

### Steady-state absorption spectra

The steady-state absorbance spectra (Fig. 1a) exhibit an absorption onset at ~1.6 eV and a monotonic increase with photon energy. These features agree with reports on the same compound^{24} and their relatives (MAPbI_{3})^{25}. Yang et al. employed a detailed model, based on Elliot’s theory^{26}, to analyze both the steady-state and transient absorption spectra, assuming both the conduction band and valence band possess parabolic dispersion^{4,27}. Thereafter, many works follow the same parabolic-band assumption in their fits to steady-state absorption, and subsequent transient absorption data analysis^{21,22}.

However, Saba et al.^{28,29} revealed that the steady-state absorption spectra should be fitted by introducing a nonparabolic factor, especially for photon energies well above the bandgap. Accurate fits to steady-state absorption spectra are crucial in analyzing subsequent transient absorption data, since the extracted fitting parameters are fixed in fitting the photoinduced spectrum change. Here, we employ Elliot’s formula as well as introducing a nonparabolic band^{26,28,29,30}, using

$$A(\hslash \omega )={A}_{{{{{{{{\rm{CC}}}}}}}}}(\hslash \omega )+{A}_{{{{{{{{\rm{EC}}}}}}}}}(\hslash \omega ),$$

(1)

where the carrier contribution (CC) term is

$${A}_{{{{{{{{\rm{CC}}}}}}}}}(\hslash \omega )={A}_{1}\frac{2\pi \sqrt{{E}_{{{{{{{{\rm{b}}}}}}}}}}}{\hslash \omega }\frac{1}{{{\Gamma }}}\int\nolimits_{{E}_{{{{{{{{\rm{g}}}}}}}}}}^{\infty }{{{{{{{\rm{se}}}}}}}}{{{{{{{\rm{ch}}}}}}}}\left(\frac{\hslash \omega -E}{{{\Gamma }}}\right)\frac{\xi (E-{E}_{{{{{{{{\rm{g}}}}}}}}})}{1-\exp \left(-2\pi \sqrt{\frac{{E}_{{{{{{{{\rm{b}}}}}}}}}}{E-{E}_{{{{{{{{\rm{g}}}}}}}}}}}\right)}dE,$$

(2)

and the exciton contribution (EC) term is

$${A}_{{{{{{{{\rm{EC}}}}}}}}}(\hslash \omega )={A}_{2}\frac{2\pi {{E}_{{{{{{{{\rm{b}}}}}}}}}}^{3/2}}{\hslash \omega }\frac{1}{{{\Gamma }}}\mathop{\sum }\limits_{j=1}^{{j}_{max}=7}\frac{1}{{j}^{3}}{{{{{{{\rm{se}}}}}}}}{{{{{{{\rm{ch}}}}}}}}\left(\frac{\hslash \omega -{E}_{{{{{{{{\rm{g}}}}}}}}}+{E}_{{{{{{{{\rm{b}}}}}}}}}/{j}^{2}}{{{\Gamma }}}\right),$$

(3)

where the prefactors *A*_{1} and *A*_{2} are proportional to the transition dipole moment, *E*_{g} is the bandgap, *E*_{b} is the exciton binding energy, (*E*_{g} − *E*_{b}/*j*^{2}) is the energy of the *j*^{th} excitonic state with *j*_{max} for cutoff (We chose *j*_{max} = 7, which is high enough to account for all the excitonic contribution to the absorption spectrum), the \({E}_{{{{{{{{\rm{b}}}}}}}}}^{3/2}\) in the EC term arises from the coupling between the wavefunctions of an electron and a hole^{31}, Γ is the exciton bandwidth and \(\xi (E-{E}_{{{{{{{{\rm{g}}}}}}}}})=1+10R(E-{E}_{{{{{{{{\rm{g}}}}}}}}})+126{R}^{2}{(E-{E}_{{{{{{{{\rm{g}}}}}}}}})}^{2}\) is to account for band nonparabolicity effect with *R* being the nonparabolic factor^{29}. Using the nonparabolic term described in Eq. (1) enables us to fit the data over a wide range of photon energies from 1.5 to 2 eV (see Fig. 1b). Without the nonparabolic term *R*, the fitting quality is poor (see Supplementary Note 1), while the spectra can be only well fit from 1.5 to 1.8 eV as shown in Fig. 1a. Including the *R* terms in the fitting model improves the fit quality, increases the energy range of fit, and also enables the fitting parameters to be stable with respect to a slight change in the fitting range (compare Supplementary Tables 1 and 2). A fit to the data up to 2.0 eV, shown in Fig. 1b, yields *E*_{g} = 1.631 ± 0.008 eV and *E*_{b} = 23 ± 5 meV. We then fix these extracted parameters in the fitting model to fit the subsequent transient absorption data.

### Determination of carrier temperature

We further measured the TA spectra on FCPIB with above-bandgap (400 nm, for 2D contour plot see Supplementary Fig. 1) and resonant (770 nm) excitation. Figure 2a and 2c show the normalized TA spectra, Δ*A*(*ℏ**ω*), at representative time delays from 0.5 ps to 1 ns under 400-nm and 770-nm pump with a fluence of 83 μJ/cm^{2}. In Fig. 2a, a photoinduced absorption (PIA) observed below the bandgap at short time delays (e.g., 0.5 ps) has been attributed to the presence of additional optical transitions due to bandgap renormalization (BGR), rather than biexciton formation^{32}. Such feature is absent in the TA spectra for 770-nm excitation. A large bleaching feature at energies above the bandgap implies excited states being occupied by photogenerated carriers and excitons. The probe photon energy dependence of the bleaching yields a photogenerated carrier energy distribution that obeys the Fermi-Dirac (FD) distribution. Under the condition of *ℏ**ω* ≫ \({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\) (quasi-Fermi energy), the carrier distribution can be simplified to the Maxwell-Boltzmann (MB) distribution^{4,21}, leading to the expression

$${{\Delta }}A(\hslash \omega )=-C\exp \left(-\frac{\hslash \omega }{{k}_{{{{{{{{\rm{B}}}}}}}}}{T}_{{{{{{{{\rm{e}}}}}}}}}}\right),$$

(4)

where *C* is a prefactor, *k*_{B} is the Boltzmann constant, and *T*_{e} is the carrier temperature. By fitting the high-energy tail of the data (see Fig. 2a) to Eq. (4), we obtain *T*_{e} as a function of pump-probe delay, as shown in Fig. 2b for different pump fluences. The 400-nm pump results show slow carrier cooling under high pump fluences that are consistent with previous works in this family of perovskite materials^{4}. In contrast, the normalized Δ*A*(*ℏ**ω*) after 770-nm excitation shows a negligible change in the carrier temperature (Fig. 2d) by fitting the data to the MB distribution (Fig. 2c). The negligible photoinduced change in the *T*_{e} for 770-nm pump (see Fig. 2d) is expected, as there is little excess kinetic energy given to the electrons once they have been excited to the conduction band. Note that the *T*_{e} in Fig. 2d is slightly higher than the room temperature value of 294 K, and is the consequence of not taking into account the transient change in the reflection^{32}.

### Transient changes of carriers and excitons

Next we analyze data over the entire spectral range to extract the other parameters, at each time delay. After photoexcitation, a transient change in the bandgap results from BGR and state-filling, which coexist and compete with each other^{33}. Here we introduce a photoinduced change in bandgap (Δ*E*_{g}) to account for BGR, and insert a \({(1-{f}_{{{{{{{{\rm{FD}}}}}}}}})}^{2}\) factor to account for state-filling effects in both the CB (for electrons) and VB (for holes), where *f*_{FD} is the Fermi-Dirac distribution function. The CC term is given by

$${{\Delta }}{A}_{{{{{{{{\rm{CC}}}}}}}}}(\hslash \omega )= \; {A}_{CC}^{\prime}({E}_{{{{{{{{\rm{b}}}}}}}}}^{\prime},{E}_{{{{{{{{\rm{g}}}}}}}}}^{\prime},\hslash \omega ){(1-{f}_{{{{{{{{\rm{FD}}}}}}}}})}^{2}-{A}_{{{{{{{{\rm{CC}}}}}}}}}(\hslash \omega )\\ = \; {f}_{1}{A}_{1}\frac{2\pi \sqrt{{E}_{{{{{{{{\rm{b}}}}}}}}}^{\prime}}}{\hslash \omega }\frac{1}{{{\Gamma }}}\int\nolimits_{{E}_{{{{{{{{\rm{g}}}}}}}}}^{\prime}}^{\infty }{{{{{{{\rm{se}}}}}}}}{{{{{{{\rm{ch}}}}}}}}\left(\frac{\hslash \omega -E}{{{\Gamma }}}\right)\frac{\xi (\hslash \omega -{E}_{{{{{{{{\rm{g}}}}}}}}}^{\prime})}{1-\exp \left(-2\pi \sqrt{\frac{{E}_{{{{{{{{\rm{b}}}}}}}}}^{\prime}}{E-{E}_{{{{{{{{\rm{g}}}}}}}}}^{\prime}}}\right)}dE\\ \times {\left(1-\frac{1}{1+\exp \left(\frac{\hslash \omega -{E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}}{{k}_{{{{{{{{\rm{B}}}}}}}}}{T}_{{{{{{{{\rm{e}}}}}}}}}}\right)}\right)}^{2}-{A}_{1}\frac{2\pi \sqrt{{E}_{{{{{{{{\rm{b}}}}}}}}}}}{\hslash \omega }\frac{1}{{{\Gamma }}}\int\nolimits_{{E}_{{{{{{{{\rm{g}}}}}}}}}}^{\infty }{{{{{{{\rm{se}}}}}}}}{{{{{{{\rm{ch}}}}}}}}\left(\frac{\hslash \omega -E}{{{\Gamma }}}\right)\\ \times \frac{\xi (\hslash \omega -{E}_{{{{{{{{\rm{g}}}}}}}}})}{1-\exp \left(-2\pi \sqrt{\frac{{E}_{{{{{{{{\rm{b}}}}}}}}}}{E-{E}_{{{{{{{{\rm{g}}}}}}}}}}}\right)}dE,$$

(5)

where \({E}_{{{{{{{{\rm{b}}}}}}}}}^{\prime}\equiv {E}_{{{{{{{{\rm{b}}}}}}}}}+{{\Delta }}{E}_{{{{{{{{\rm{b}}}}}}}}}\), \({E}_{{{{{{{{\rm{g}}}}}}}}}^{\prime}\equiv {E}_{{{{{{{{\rm{g}}}}}}}}}+{{\Delta }}{E}_{{{{{{{{\rm{g}}}}}}}}}\), *f*_{1} accounts for the photoinduced change in the transition dipole moment embedded in *A*_{1} (*f*_{1} is found to be ~1 after pump in our fitting, so we fix *f*_{1} = 1), and \({(1-{f}_{{{{{{{{\rm{FD}}}}}}}}})}^{2}\) accounts for occupied transitions by carriers, due to the Pauli exclusion principle and characterized by the quasi-Fermi energy \({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\) and the carrier temperature *T*_{e}. Equation (5) assumes (1) the electron and hole effective masses are nearly the same, (2) a negligible photoinduced change in the nonparabolic parameter *R*, and (3) *T*_{e} ≈ *T*_{h}, where *T*_{h} is the temperature of holes.

Next we consider the EC term. Considering the changes in *E*_{b}, Γ, and *E*_{g} after photoexcitation, the photoinduced change in the EC term can be expressed as

$${{\Delta }}{A}_{{{{{{{{\rm{EC}}}}}}}}}(\hslash \omega )= \; {f}_{2}{A}_{EC}^{\prime}({E}_{{{{{{{{\rm{b}}}}}}}}}^{\prime},{E}_{{{{{{{{\rm{g}}}}}}}}}^{\prime},{{{\Gamma }}}^{\prime},\hslash \omega )-{A}_{{{{{{{{\rm{EC}}}}}}}}}(\hslash \omega )\\ = \; {f}_{2}{A}_{2}\frac{2\pi \sqrt{{E}_{{{{{{{{\rm{b}}}}}}}}}^{\prime}}}{\hslash \omega }\frac{1}{{{{\Gamma }}}^{\prime}}\mathop{\sum }\limits_{j=1}^{7}\frac{2{E}_{b}^{\prime}}{{j}^{3}}{{{{{{{\rm{se}}}}}}}}{{{{{{{\rm{ch}}}}}}}}\left(\frac{\hslash \omega -{E}_{{{{{{{{\rm{g}}}}}}}}}^{\prime}+\frac{{E}_{{{{{{{{\rm{b}}}}}}}}}^{\prime}}{{j}^{2}}}{{{{\Gamma }}}^{\prime}}\right)\\ -{A}_{2}\frac{2\pi \sqrt{{E}_{{{{{{{{\rm{b}}}}}}}}}}}{\hslash \omega }\frac{1}{{{\Gamma }}}\mathop{\sum }\limits_{j=1}^{7}\frac{2{E}_{b}}{{j}^{3}}{{{{{{{\rm{se}}}}}}}}{{{{{{{\rm{ch}}}}}}}}\left(\frac{\hslash \omega -{E}_{{{{{{{{\rm{g}}}}}}}}}+\frac{{E}_{{{{{{{{\rm{b}}}}}}}}}}{{j}^{2}}}{{{\Gamma }}}\right),$$

(6)

where \({{{\Gamma }}}^{\prime}\equiv {{\Gamma }}+{{\Delta }}{{\Gamma }}\), and *f*_{2} is the prefactor that accounts for the photoinduced change in exciton oscillator strength^{31}, which we will show later to be a strong function of pump fluence. The prefactors *A*_{1} and *A*_{2} are fixed values that are obtained from the fitting of the steady-state absorption spectrum earlier, as given in Eq. (2) and Eq. (3).

Combining both the CC and EC terms, we then fit the TA data Δ*A*(*ℏ**ω*) to

$${{\Delta }}A(\hslash \omega )={{\Delta }}{A}_{{{{{{{{\rm{CC}}}}}}}}}(\hslash \omega )+{{\Delta }}{A}_{{{{{{{{\rm{EC}}}}}}}}}(\hslash \omega ).$$

(7)

During the fitting, we found that \({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\) and *T*_{e} are dependent on each other, i.e., we can find different sets of (\({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\), *T*_{e}) values that can fit the data equally well (see Supplementary Note 2). This is a consequence of the fact that *T*_{e} and \({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\) appear in the FD term, and *only* the FD term, via the exponent in Eq. (5), and that \({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\) will adjust itself depending on the value of *T*_{e}, to yield identical fits. To overcome the problem, since we have obtained *T*_{e} from the MB fits to the normalized Δ*A*(*ℏ**ω*) at high photon energies, we fix *T*_{e} (extracted from the MB fits in Eq. (4)) to obtain \({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\) in the fits to Eq. (7). In Eq. (7), by fixing the parameters *T*_{e}, *E*_{g}, *E*_{b}, and Γ, the fitting parameters are therefore Δ*E*_{g}, Δ*E*_{b}, ΔΓ, \({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\) and *f*_{2}. We then plot Δ*A*_{CC}(*ℏ**ω*) and Δ*A*_{EC}(*ℏ**ω*), separately, to visualize the carrier (CC) and exciton (EC) terms at each time delay.

Figure 3 shows the CC and EC terms under 400-nm excitation at selected time delays. At 0.5 ps, the bleaching shows a sharp onset at 1.63 eV and a very long high-energy tail. From Fig. 3b, c, we see that the sharp bleaching onset is contributed by both CC and EC terms, while the high-energy tail can be attributed to the state-filling of hot carriers described by the FD distribution. After several picoseconds, the bleaching dip contributed by carriers moves towards lower-energy states as the carriers lose energy while cascading down to the bandedge, ie. carrier cooling. At the same time, the carrier bleaching amplitude near the bandedge becomes larger, suggesting the predominance of bandege transitions (to both carrier and excitonic levels), which is the result of a large number of carriers near the bandedge. At long time delays (Fig. 3e), as carriers keep cooling down to bandedge, the bleaching contributed by carriers moves to even lower energies in the band but the bleaching amplitude decreases with time delay, indicative of a smaller carrier number due to carrier recombination. The EC term does not merely show the change in amplitude, but also in the linewidth and energies of lowest excitonic transitions, which we can recognize from the dip position, as shown in Fig. 3c and f.

### Quasi-Fermi energy and unusual bandgap renormalization

From the Δ*A* fittings, we extracted the quasi-Fermi energy \({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\) under 400-nm and 770-nm photoexcitation, as shown in Fig. 4(a) and (b), respectively. For 400-nm excitation (Fig. 4(a)), \({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\) increases from ~1.4 to 1.6 eV within the first ~2 ps. This increase is consistent with more electrons being promoted to the conduction band. Thereafter, \({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\) decreases slowly to ~1.55 eV at 1 ns, suggesting that even after 1 ns, there is still a significant electron population at the conduction band minimum. On the other hand, for 770-nm excitation shown in Fig. 4(b), the extracted \({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\) remains at ~1.52 eV throughout (see Supplementary Note 3 for detailed fittings). This is consistent with the very fast (<0.5 ps timescale) and resonant absorption of the pump photons by the valence band electrons to the conduction band minimum.

In traditional semiconductors, such as a heavily Sn-doped In_{2}O_{3}, the optical bandgap increases with carrier density *N*^{34}. This bandgap widening is the combined effect of Burstein-Moss effect and bandgap renormalization, in which the latter contributes to bandgap reduction^{34,35}. Walsh et al. later revealed that the nonparabolic nature of the conduction band is a key factor of this renormalization^{33}. In inorganic semiconductors such as ZnO nanocrystals, the bandgap decreases with increasing lattice temperature^{36}. However, lead halide perovskites exhibit an atypical temperature dependence of bandgap—the bandgap increases with increasing temperature^{20,37}. This temperature dependence of the bandgap in MAPbI_{3} is attributed to both thermal expansion and electron–phonon interactions, of which the former predominantly contributes to the BGR^{38}. This temperature dependence of the fundamental gap makes the sign of photoinduced change in bandgap uncertain—it is not entirely certain that BGR necessarily results in a bandgap reduction in the perovskites.

Figure 4(c) and (d) show the time evolution of Δ*E*_{g} under 400-nm pump and 770-nm pump. In contrast to previous TA works^{4,21,22}, our extracted Δ*E*_{g} is positive at all time delays for all pump fluences, implying that the bandgap increases after photoexcitation. This is consistent with the temperature dependence of the bandgap^{20,37}.

### Carrier and exciton dynamics

After photoexcitation, part of the conduction band (valence band) is occupied by photoexcited electrons (holes). These filled states lead to ground-state bleaching. In past reports, the time evolution of the carrier density was obtained via time-resolved terahertz spectroscopy (TRTS)^{39,40}. Here we show that TA data can do the same, and thus serve as a complimentary check to the TRTS results. With the assumption that the electron density is equal to the hole density, the electron density *N*_{e} can be a measure of carrier density *N*, that is calculated from the extracted fitting parameters *E*_{g}, *E*_{b}, \({E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}}\), and *T*_{e} via the formula \({N}_{{{{{{{{\rm{e}}}}}}}}}=\int\nolimits_{{E}_{{{{{{{{\rm{g}}}}}}}}}}^{\infty }{D}_{c}(\hslash \omega ){f}_{{{{{{{{\rm{FD}}}}}}}}}d\hslash \omega\), where *D*_{c}(*ℏ**ω*) is the density of states of conduction band^{4}. The full expression is

$${N}_{{{{{{{{\rm{e}}}}}}}}}=\int\nolimits_{{E}_{{{{{{{{\rm{g}}}}}}}}}}^{\infty }\frac{1}{2{\pi }^{2}}{\left(\frac{2{m}_{{{{{{{{\rm{eff}}}}}}}}}}{{\hslash }^{2}}\right)}^{\frac{3}{2}}\sqrt{E-{E}_{{{{{{{{\rm{g}}}}}}}}}}\frac{\xi (E-{E}_{{{{{{{{\rm{g}}}}}}}}})}{1+\exp ((E-{E}_{{{{{{{{\rm{f}}}}}}}}}^{{{{{{{{\rm{q}}}}}}}}})/{k}_{{{{{{{{\rm{B}}}}}}}}}T)}dE$$

(8)

Figure 5a shows the obtained time evolution of electron densities *N*_{e} for different 400-nm pump fluences. The higest pump flunece of 83 μJ/cm^{2} yields a carrier density on the order of 10^{20} cm^{−3} at short time delay. The values of *N*_{e} at smaller pump fluences have the same order of magnitude as that of CH_{3}NH_{3}PbI_{3} under 400-nm pump and similar pump fluences using time-resolved THz spectroscopy^{40}. By modeling the decay of the electron densities *N*_{e} by the rate equation given by −*d**N*(*t*)/*d**t* = *k*_{1}*N* + *k*_{2}*N*^{2} + *k*_{3}*N*^{3}, we identified the carrier recombination process for different pump fluences (see Supplementary Note 4) and found a pump-fluence-dependent carrier recombination mechanism, that is, the carrier recombination process undergoes a transition from second-order for high-fluence excitation (≥56 μJ/cm^{2}) to third-order for low-fluence excitation (≤42 μJ/cm^{2}). The insets of Fig. 5b, c and d show the fitting curves of 83 μJ/cm^{2} for a second-order process and of 42 and 14 μJ/cm^{2} for a third-order process in the range of 20–1000 ps. A change of recombination kinetics has been previously reported in CH_{3}NH_{3}PbI_{3} perovskite film, that is from two-body recombination at room temperature to three-body recombination at low temperatures^{40}. Though not in the sense of lattice temperature, the higher pump fluences yield higher carrier temperatures, which might explain the similar change of the recombination kinetics for pump fluence and temperature. The extrapolated fitting curve to shorter time delays (see Fig. 5b) underestimates the experimental electron density. This underestimation was also observed in a time-resolved THz spectroscopy work of FCPIB^{24}. This discrepancy suggests the existence of additional decay channels for the carriers at early time delays, which we attribute to exciton formation, as will be justified later in our discussion.

After obtaining the carrier and exciton dynamics from our fits, we now provide a physical picture of what happens to the photogenerated carriers, as shown in Fig. 6. After 400-nm photoexcitation, electrons are generated in the CB. These electrons then cascade down to the bottom of the CB via electron–phonon interactions. At the CB minimum, there are two possible paths for the electrons to take: (1) some electrons recombine with holes in the VB (see Fig. 6a) via radiative or non-radiative recombination—the recombination order (second or third) depends on the pump fluence; (2) other electrons bind with holes, forming excitons in the lower-lying excitonic levels via phonon emission (see Fig. 6b). Path (2) accounts for the discrepancy between the experimental *N*_{e} and theoretical *N*_{e} extrapolated from the carrier recombination curve (solid lines in Fig. 5b–d).

The physical picture above is substantiated by the time evolution of the exciton oscillator strength *f*_{2}. Strictly speaking, *f*_{2} is the ratio of exciton oscillator strength after photoexcitation to that before photoexcitation, and so is a measure of the excitonic state-filling effect. The (bound) electron and hole levels that are already filled will not be available for subsequent electrons in the CB (and holes in the VB) to fill them, so a small *f*_{2} implies a large excitonic state-filling effect. Taking the highest-fluence data of 83 μJ/cm^{2} as an example, at the shortest time delays (<1 ps), *f*_{2} is already 0.25 (see Fig. 6d), i.e. much less than 1, indicating a rapid filling of the excitonic states (see region *i* in Fig. 6d) and a concurrent rapid decay of *N*_{e} (see Fig. 5a). Up till ~10 ps, *f*_{2} continues to decrease, resulting from more excitonic states being filled, and reaches a minimum value of ~0.1 at ~10 ps (see region *i**i* in Fig. 6d). At ~10 ps, *f*_{2} starts to increase, corresponding to the gradual “vacating” of the previously-filled excitonic states either via excitonic recombination or excitonic dissociation, while carrier recombination becomes more and more dominant (see region *i**i**i* in Fig. 6d). From ~10 ps to ~1 ns, the quenched *f*_{2} starts to recover and approaches unity. At ~1 ns, the other fitting parameters also return to their pre-photoexcitation values (see Supplementary Fig. 9).

At lower pump fluences (e.g., 14 and 28 μJ/cm^{2} in Fig. 6c), *f*_{2} is closer to 1, and stays at a constant value for a longer period of time after photoexcitation. The corresponding *N*_{e} is smaller in magnitude, and does not manifest a sharp upturn at the shortest time delays (see Fig. 5a). Nonetheless the general trend persists. Our physical picture of the fate of the photogenerated carriers are thus confirmed by the time evolution of *N*_{e} (with its rate equation fits) and *f*_{2}. The very long (nanosecond) timescale of our carrier dynamics is also consistent with polaron formation as reported in other reports^{24,41,42}. At high pump fluences and short time delays, the excitons are more strongly depleted, which suggests the use of a high pump fluence as a knob to tune the exciton populations and hence the carrier-to-exciton conversion efficiencies. Since the binding energy of the excitons has to be overcome in order to obtain free charge carriers and to make the current flow, a higher pump fluence below the damage threshold may benefit the performance of perovskites in photovoltaic applications.