Hot carrier cooling mechanisms in halide perovskites

Halide perovskites exhibit unique slow hot-carrier cooling properties capable of unlocking disruptive perovskite photon–electron conversion technologies (e.g., high-efficiency hot-carrier photovoltaics, photo-catalysis, and photodetectors). Presently, the origins and mechanisms of this retardation remain highly contentious (e.g., large polarons, hot-phonon bottleneck, acoustical–optical phonon upconversion etc.). Here, we investigate the fluence-dependent hot-carrier dynamics in methylammonium lead triiodide using transient absorption spectroscopy, and correlate with theoretical modeling and first-principles calculations. At moderate carrier concentrations (around 1018 cm−3), carrier cooling is mediated by polar Fröhlich electron–phonon interactions through zone-center delayed longitudinal optical phonon emissions (i.e., with phonon lifetime τ LO around 0.6 ± 0.1 ps) induced by the hot-phonon bottleneck. The hot-phonon effect arises from the suppression of the Klemens relaxation pathway essential for longitudinal optical phonon decay. At high carrier concentrations (around 1019 cm−3), Auger heating further reduces the cooling rates. Our study unravels the intricate interplay between the hot-phonon bottleneck and Auger heating effects on carrier cooling, which will resolve the existing controversy.

. Note that the TA data were obtained from our Helios setup with a 750 nm short pass filter placed in the probe path before the sample 1 . The absence of a 750 nm short pass filter to eliminate the residual 800 nm in the white light probe will severely influence the results obtained (See Supplementary Figure 19). Therefore, due care must be taken to ensure that such filter is used when collecting the TA spectra.  As discussed in our earlier work 17 , the measured hot-phonon lifetime could also be limited by the timeresolution of the experimental techniques used, thereby yielding artificially longer lifetimes that are limited by the system temporal response rather than its intrinsic hot-phonon lifetime. Hence, due care must be taken for a fair comparison of the reported values in the literature.

Supplementary Note 1. Analysis of absorption spectra
The absorption coefficient of the sample is calculated using the following equation 18 where g E is the bandgap, µ is the reduced mass, 0 m is the free electron mass, % n is the refractive index, cv f is the oscillator strength, ( ) x θ is the unit step function, * 0 R is the effective Rydberg.
The estimated bandgap and the exciton binding energy are respectively 1.651±0.001 eV and 6.3±0.1 meV, which is in agreement with the measured value using high magnetic fields 21 . The negligible binding energy as well as insignificant exciton contribution to the band edge absorption indicates a Wannier-type exciton in MAPbI 3 films and immediate dissociation of exciton into free carriers after photoexcitation, which to some degree explains the excellent performance of lead iodide perovskite MAPbI 3 in photovoltaic applications.

Supplementary Note 2. Estimation of trap density
The carrier density is estimated by 0 , which is exceptionally low for solution prepared films -indicating the excellent properties of MAPbI 3 for optoelectronic applications.  23 where h is the reduced Planck's constant,

Supplementary Note 3. Model for band gap shifting
is the reduced mass, and n is the carrier density. The contribution from the reduction of free exciton binding energy can be described by: . The band gap of the semiconductor after photoexcitation with all the contributions taken into account is:

Supplementary Note 4. Analysis of TA spectra and hot carrier distribution
From Fermi's golden rule, the absorption coefficient for a direct semiconductor within a parabolic band approximation is: where cv C is a constant related with transition matrix, is the joint density of states with the band gap of g where 0 ( ) A E is steady state linear absorbance. One therefore will always obtain a negative A Δ The estimated monomolecular recombination coefficient is where k 3 is the Auger coefficient. We global fit the A Δ kinetics obtained at different carrier densities. The monomolecular recombination rate 1 k was obtained from TRPL measurement (Supplementary Figure 12 (c)). The estimated bimolecular recombination coefficient and Auger coefficient are respectively  Figure 13). Compared to the hot phonon effect, influence of free carrier screening effect has a negligible effect on HC cooling. Therefore, we conclude here that slow HC cooling in halide perovskites is largely unaffected by free carrier Coulomb screening effect over the carrier densities of 10 17 -10 19 cm -3 .
perovskites APbI 3 (A can be CH 3 NH 3 , NH 2 CH=NH 2 and Cs), which also leads to slow hot hole cooling.
In light of the relation between the effective mass and the hot carrier cooling rate, the similarity of perovskite's electron and hole effective masses aptly highlights its distinct advantage over conventional semiconductors for hot carrier applications. In typical semiconductors, their electron effective mass is much lower than that of the hole. This will result in faster hot hole cooling compared to hot electron cooling. In contrast, the more "balanced" slow hot electron and hot hole cooling in perovskites will be more amenable for developing practical hot-carrier optoelectronic devices.

Supplementary Note 9. Calculated Phonon spectra and projected density of states
Using the first-principles method based on density functional theory, we calculated the phonon dispersion spectra and the corresponding projected density of states (pDOS) of the MAPbI 3 in the tetragonal and orthorhombic phases, which are shown in Supplementary Figure 14. We find that both phonon spectra for the two phases are similar in low frequency (less than 16 meV).