Confinement and Exciton Binding Energy Effects on Hot Carrier Cooling in Lead Halide Perovskite Nanomaterials

The relaxation of the above-gap (“hot”) carriers in lead halide perovskites (LHPs) is important for applications in photovoltaics and offers insights into carrier–carrier and carrier–phonon interactions. However, the role of quantum confinement in the hot carrier dynamics of nanosystems is still disputed. Here, we devise a single approach, ultrafast pump–push–probe spectroscopy, to study carrier cooling in six different size-controlled LHP nanomaterials. In cuboidal nanocrystals, we observe only a weak size effect on the cooling dynamics. In contrast, two-dimensional systems show suppression of the hot phonon bottleneck effect common in bulk perovskites. The proposed kinetic model describes the intrinsic and density-dependent cooling times accurately in all studied perovskite systems using only carrier–carrier, carrier–phonon, and excitonic coupling constants. This highlights the impact of exciton formation on carrier cooling and promotes dimensional confinement as a tool for engineering carrier–phonon and carrier–carrier interactions in LHP optoelectronic materials.

: To extract the cooling time constants from the push-induced PPP dynamics, all data are fitted with a convolution of a Gaussian and exponential function. The bleach rise is most accurately modelled when σ = 100 fs and is used for all data sets.

Figure S3:
The absorption spectra of the studied materials were modelled using Elliott theory described in Section C below. The more confined systems display a larger exciton binding energy and further excitonic transitions above the band gap.     TOP-Br2 precursor, 0.5 M in toluene: TOP (6 ml, 13 mmol) and bromine (0.6 ml, 11.5 mmol) were mixed under inert atmosphere. Once the reaction was complete and cooled to room temperature, the TOP-Br2 was dissolved in toluene (18.7 ml).

Synthesis of CsPbBr3
NCs with ASC18 ligand: CsPbBr3 NCs were synthesised by dissolving Csoleate (4 ml, 1.6 mmol), Pb-oleate (5 ml, 2.5 mmol) and ASC18 (0.215 g, 0.512 mmol) in ODE (10 ml) and heating the mixture to 130 °C under vacuum, whereupon the atmosphere was changed to argon and TOP-Br2 in toluene (5 ml, 5 mmol Br) was injected. The reaction was cooled immediately in an ice bath.
The crude solution (24 ml) was precipitated by the addition of ethyl acetate (20 ml) and acetone (21 ml) in a nitrogen-filled glovebox, followed by centrifugation at 29500g for 10 min. The precipitated fraction was dispersed in toluene (3 ml) and then washed three more times. Each time the solution was mixed with two volumetric equivalents of acetone and centrifuged at 1300g for 10 min, before being dispersed in progressively smaller volumes of solvent (1.5 ml, then 0.75 ml). After the final precipitation, NCs were dispersed in toluene (3 ml) and centrifuged at 1300g for 1 min to remove non-dispersed residue.
CsPbBr3 NC films: Z-cut single crystalline quartz substrates were cleaned by sonication in soap water and water stream, then blow dried (sequence repeated twice). They were then sonicated in ethanol, and separately in acetoneblow-drying between stepsand covered with a monolayer of hexamethyldisilazane before annealing in a nitrogen-filled glovebox. Films were prepared by dropcasting a 0.8 mg/ml NC solution. flow. When 130 °C was reached, 0.8 ml Cs-oleate prepared as described above was swiftly injected and the reaction was immediately stopped by immersing the reaction flask into a cold-water bath. After the reaction, 1 ml of crude solution was centrifuged for 3 minutes at 5000 rpm. The obtained precipitate was dispersed in 1 ml toluene, centrifuged again for 10 minutes at 13400 rpm, and the supernatant was filtrated and used for the experiments.

Ultrafast spectroscopy
A Ti:sapphire regenerative amplifier (Astrella, Coherent, λc ~ 800 nm, τ ~ 35 fs) seeded two optical parametric amplifiers (TOPAS-Prime, Coherent) to produce near-infrared light. The output of "OPA-1" was tuned to ~1200 nm and coupled into a β-barium borate crystal alongside residual 800 nm light, producing the 490 nm "pump", which was then modulated at 2 kHz by an optical chopper. The output of "OPA-2" was tuned to 2 µm, where it was then split into two paths by a beamsplitter, with 90% of the incident light forming the "push" and the remaining 10% the "probe". Pump and probe beams were sent into separate mechanical delay stages: the pump-push delay time was controlled through the position of the pump stage, which was fixed for all pump-push-probe measurements at ~12 ps. The pump-probe delay time was scanned by moving the probe stage. The pump and push fluences were controlled using a neutral density filter wheel. The pump and probe then adopted a collinear geometry and were focused onto a ~200 µm diameter spot on the sample housed in a N2-filled quartz cuvette; the off-axis push was defocused to ~400 µm to reduce photodegradation and aid spatial overlap. The transmission of the probe was detected by an amplified PbSe photodetector (PDA20H-EC, Thorlabs), and the differential signal read out by a lock-in amplifier (MFLI, Zurich Instruments).

B. Monte Carlo simulation for cooling within NCs
To demonstrate the validity of assuming an effectively bulk-like carrier density throughout the studied systems, we have modelled the push pulse action on a test set of confined NCs using a Monte Carlo algorithm. Here, ℎν push photons are randomly assigned to 10000 NCs. With increasing ℎν , the probability distribution of forming 0, 1 or 2 hot carriers in each NC shifts to higher values. If we assume that cooling within a singly occupied NC takes 1 , and cooling in a doubly occupied NC takes 2 (where  This result obtained from a strictly confined system is very similar to that predicted by a bulk free carrier model. 1,2 As such, both classes of system and their intermediates can be described by the same model as used in the manuscript.

C. Fitting of absorption spectra with Elliott model
To verify the exciton binding energy of each studied material, we apply an Elliott model to the absorption spectra, which are decomposed into a linear combination of contributions by exciton ( x ) and continuum ( c ) states: The sub-gap exciton states are modelled in the first term of (2) by a series of delta functions at = g − b 2 (so converges at g ), with intensities that scale with 1/ 3 . In practice, ≲ 5 is typically sufficient.
The second term of (2) describes an exciton-like transition found above the band gap at = 2 in the more confined systems.
Above g , the continuum scales with √ − g , which assumes a parabolic band structure near the band edge.
x , x and c are scaling factors to fit the model to the experimental data.
Line broadening due to static and dynamic disorder is modelled by convolving the exciton and continuum contributions with Gaussian functions. The final fitted spectra thus have the form: The results for each material are plotted in Figure S2 and show good agreement with the absorption spectra using b values similar to those reported previously. [3][4][5][6] It has been shown that the absorption profile of bulk perovskites at the band edge is sensitive to fabrication procedures which cause exciton features to be more pronounced despite a relatively low b in these systems. 7 The value used is therefore based on the literature and not verified further in this work. In the case of the (PEA)2PbI4 absorption spectrum, subtraction of a spline describing the profile between the exciton and continuum contributions yields a better fit to the model.

D. Kinetic model for the effect of exciton binding energy on hot carrier cooling dynamics
To gain further insight into the effect of the materials' excitonic character on their hot carrier cooling dynamics, we have developed a numerical kinetic model based on a system of coupled differential equations: These rate equations are based on the following assumptions: • hot = HC + HX , where HC is the hot (free) carrier density and HX is the hot exciton density; • Excited states may be either hot or cold; • HCs and HXs become cold with identical efficiencies via two mechanisms: (i) energy deposition into vacant phonons (with density ph ); and (ii) scattering with the reservoir of cold states (with density cold ); • An additional cooling pathway is available to the HXs; • Occupied phonon modes may not contribute to cooling; • , and are the rate constants for hot-cold carrier (or carrier-exciton, exciton-exciton) scattering, carrier-phonon scattering and hot exciton cooling, respectively; • Band-to-band recombination and occupied phonon freeing rates are negligible on timescales relevant for hot carrier cooling.
We note that under relatively soft assumptions, equation (3) can be excluded and equation (1) takes the form: The total phonon density is represented by ph , the carrier-phonon interaction (polaron) volume by ′ , and Δ hot is the change in hot state density after the push pulse. This equation is given in the main text of the paper.
The following initial conditions are imposed for the simulations: • hot (0) = push × pump ; • cold (0) = (1 − push ) × pump ; where 0 < push < 1 and describes the intensity of the push pulse and thus the optically re-excited portion of the total (cold) excited state density formed by the pump ( pump ). The push is modelled as being instantaneous for simplicity; • ph (0) = ph , the total number of phonon modes, i.e. all phonon modes are assumed empty at the instant hot carriers are formed by the push.
The hot carrier-exciton partition is based on a modified Saha equation: 8,9 where = HC / hot , = ℎ/√2 B , b is the material's exciton binding energy, and is temperature. We assume a reduced electron-hole mass of = 0.2 e for all materials, and do not attempt to account for polaron or trap effects here. 10,11 The system of rate equations can then be expressed as:  (7) where is derived by rearranging (4) Note that the quadratic equation in (9) has exactly one positive real solution, which is taken as the true value of .
The Euler method is employed to approximate the solutions to the differential equations and simulate the time-dependent population dynamics. For example, the change in hot from a time point ( − 1) to the next ( ) is described by: That is, the change in hot between any two time points is linear, which is a reasonable approximation for sufficiently small time steps. cool is extracted by fitting the simulated hot dynamics with an exponential function of the form ( ) ∝ − / cool .