Coupling to octahedral tilts in halide perovskite nanocrystals induces phonon-mediated attractive interactions between excitons

Understanding the origin of electron–phonon coupling in lead halide perovskites is key to interpreting and leveraging their optical and electronic properties. Here we show that photoexcitation drives a reduction of the lead–halide–lead bond angles, a result of deformation potential coupling to low-energy optical phonons. We accomplish this by performing femtosecond-resolved, optical-pump–electron-diffraction-probe measurements to quantify the lattice reorganization occurring as a result of photoexcitation in nanocrystals of FAPbBr3. Our results indicate a stronger coupling in FAPbBr3 than CsPbBr3. We attribute the enhanced coupling in FAPbBr3 to its disordered crystal structure, which persists down to cryogenic temperatures. We find the reorganizations induced by each exciton in a multi-excitonic state constructively interfere, giving rise to a coupling strength that scales quadratically with the exciton number. This superlinear scaling induces phonon-mediated attractive interactions between excitations in lead halide perovskites.


S22 References. S23
Additional measured differential scattering maps are shown in Fig. S1.As discussed in the main text, we extract the timescale for the onset ( " ) and relaxation ( # ) of the lattice reorganization upon excitation by fitting the function ~exp [−/ " ] − exp [−/ # ] to the differential scattering.We fit to a weighted differential scattering intensity integrated over the entire measured q range, Δ 0 () = − ∫  7Δ(, )9(), (S1) using a q dependent weighting of () ≡ ∫ Δ(, ).The computed Δ 0 () and fits for  " and  # are shown in Fig. S2 for all pump fluences and temperatures for which time scans were measured.
In Fig. S3 we plot the extracted differential scattering as a function of the azimuthal angle on the detector, from which an isotropic response of the lattice is evident.We therefore conclude that the observed lattice response is independent of the polarization of the pump beam.

Supplementary Note 2: Thermal Response of FAPbBr3
It is important to rule out a transient heating of the NCs as a cause of the observed lattice response, as previously observed in CsPbBr3 NCs. 1 In the experiments the FAPbBr3 NCs are pumped with 400 nm photons, ~500 meV above the bandgap of the NCs, which enables the pumping of large exciton densities on the NCs.Initial carrier thermalization on sub-ps timescales will increase the temperature of the NCs (~22% of the final temperature increase from the excess ~650 meV energy), which is followed by heating stemming from the thermalization of Auger-excited hot-carriers majority (~78% of the temperature increase from the ~2.35 eV bandgap of the NCs). 2 At the highest fluence measured (0.8 mJ/cm 2 ), we estimate an average of ~50 excitons in the NC based on previously published estimates. 3Using the reported 4 room temperature heat capacity for FAPbBr3 of 1.69 MJ/m 3 K and assuming the entire pumped energy, 50x3 eV = 150 eV, is converted to heat in the 9.5 nm NCs, we estimate a maximum temperature increase of ~17 K.We performed temperature-dependent equilibrium measurements of the scattering of the FAPbBr3 NCs on the same samples with the MeV-UED instrument.In Fig. S4 we plot the differential scattering map (relative to the scattering at 100 K) as a function of temperature, Δ(, ) ≡ (, ) − (100, ).Heating from 100 to 280 K, we do see in the differential scattering some thermal expansion of the lattice, as well as a ~0.05% decrease in the 211 peak intensity.Critically, the maximum differential signal generated by a 180 K change in temperature (~0.1%) is over one order of magnitude smaller than the differential signals measured in the photo-response of the lattice.Considering we only expect at most a ~17 K increase in temperature in the photoexcited samples, the lattice thermalization response is completely negligible.Additionally, we can also argue against thermalization induced effects from the extracted timescales of the lattice response.A majority of heating induced effects should set in on a time scale of the multi-exciton decay rate (Auger-heating), in stark contrast to our observation that the lattice reorganization decreases on this timescale.

Figure S4
. Differential scattering map as a function of temperature, comparing the scattering at temperature T to the scattering at 100 K. Moderate lattice expansion is observed at elevated temperatures through a shifting to lower q of the main Bragg peaks, along with a ~0.05% decrease in the magnitude of the 211 peak (~ 2.6 Å -1 ).

The Debye Scattering Equation (DSE) Method
The DSE provides the average differential cross section (or the powder diffraction pattern) of a randomly oriented powder from the distribution of interatomic distances between atomic pairs, without any assumption of periodicity and order: where q = 4πsinθ/λ is the magnitude of the scattering vector, λ is the radiation wavelength, fi is the atomic form factor of element i, dij is the interatomic distance between atoms i and j, N is the total number of atoms and T and oi are the thermal atomic displacement parameter and the site occupancy factor associated to each atomic species, respectively.The first summation in the above equation includes the contributions of zero distances between one atom and itself and the second term (the interference term) the non-zero interatomic distances dij = | ri -rj |.This approach, compared to the conventional Rietveld method, takes advantage of the simultaneous modelling of Bragg and diffuse scattering and thus enables an appropriate description of the total scattering of samples when a limited extension of the coherent domains, the occurrence of structural defects and various kinds of compositional disorders are present, within an approach nearly free of phenomenological components.Complete DSE computation and analysis protocols have been implemented in an open source program package (the DebUsSy program Suite), 7 in which, thanks to computational tricks, such as reducing the number of interatomic distances by fully exploiting the crystal symmetry and using sampled interatomic distances, this computationally heavy approach becomes feasible.

Figure S5. a) Synchrotron X-ray total scattering data collected at 300K (black dots) and the
Rietveld fit (red trace) for ∼9 nm FaPbBr3 nanocrystals, obtained by using a split-cubic structural mode., 8 The blue line is the polynomial background implemented to model the nanocrystals diffuse scattering and some residual scattering from the He-cryostat shaft support used for the low temperature data collection (Figure 3  Atomistic models of FAPbBr3 were built using cubic-shaped nanocrystals (NCs).A cubic unit cell parameter of 6.0756 Å, determined from experimental 100 K UED data of FAPbBr3 NCs, was used for all simulations, as well as a cube-edge length of 7.1 nm (for a 8.8 nm equivalent spherical diameter).All DSE simulations were performed using sampled interatomic distances, encoded in the atomistic models, and using electron atomic scattering factors calculated by Mott-Bethe formulae.To account for the (low) angular resolution of UED experimental data, a pseudo-Voigt function was convoluted to the computed DSE patterns, using the following parameters: for the angle dependent full width at half maximum, fwhm(θ) = 0.6000 + 0.7062 tan(θ) + 0.6788 sec(θ); for the pseudo-Voigt mixing parameter, a constant η = 0.70 value.

Split-Cubic model for the FAPbBr3 structure
As discussed in the main text and in Protesescu, L. et al. 8 , FAPbBr3 NCs exhibit a disordered structure with an average-cubic phase.In this disordered phase only local distortions of the Pb-Br sublattice are expected with no long-range order, e.g.locally tilted Pb-Br-Pb bonds.The complete structural reconstruction of glassy structures from scattering data are out of reach, and simplified structural models are often employed to interpret scattering results.
X-ray total scattering measurements on the FAPbBr3 NCs, presented in Fig. 4 of the main text and in Fig. S5, lack any superstructure peaks beyond the peaks present for the cubic Pm-3m perovskite structure over the entire measured temperature range (30-300 K).The assumption of a simple Pm-3m perovskite structure, however, poorly reproduces the measured relative peak intensities.Weconclude , therefore, that the NCs exhibit a disordered phase, which, while on average cubic, has local distortions and multiple self-excluding positions of the Br ions.
This average-cubic disordered phase of perovskites has previously been described with the "split-cubic" structure model, presented, for example, in the seminal work of Protesescu, L. et al 8 for FAPbBr3 NCs 8 and, for bulk powder, in the work of Hanusch et al. 9 In this model, Br anions, linked to two Pb ions within the 3D framework built by corner-sharing PbBr6 octahedra, are disordered in four equivalent positions (with s.o.f.= 1/4, see Fig. S6a); these positions are displaced crosswise in directions orthogonal to the Pb … Pb edge, which nominally coincides with the straight Pb-Br-Pb atomic sequence of the archetypal undistorted Pm-3m cubic phase.With these displaced Br positions, the Pb-Br-Pb bond angles significantly deviate from the ideal 180° by j° (180-j being the resulting bending) The split cubic model alone provides an excellent fit to the measured diffraction pattern, as shown in Fig. S5.

Models used to simulate differential scattering
To analyze the measured differential scattering profiles (Fig. 2b and 4a), we move beyond the split cubic model, and consider all possible local distortions of the Pb-Br sublattice.We separate the local distortions into three classes of local displacements: 1) Displacements of the Br ions in directions perpendicular to the Pb-Br-Pb bonds (as in the split cubic model), inducing a tilt φ, or, equivalently, a Pb-Br-Pb bond angle.2) Displacements of the Br ions in directions parallel to the Pb-Br-Pb bonds, labelled DPb-Br.3) Displacements of the lead ions from the nominal positions ∆ TU .
We assume all such displacements are local, and while the displacements of each ion in each case are assumed to be of equal magnitude, the direction of the displacements is chosen stochastically (see figure S6b).In the data analysis presented here, we restrict displacements to the (x,0,0), (0,y,0), or (0,0,z) directions.

Correspondence between the 211 intensity changes and Pb-Br-Pb bending
In figure S6c we plot the differential diffraction profile measured with UED compared to the simulated differential profile assuming a photoexcited decrease in the Pb-Br-Pb bond angle (with increasing φ) There is a strong enhancement of the 211 peak as φ increases.
In figure S6d we plot the simulated differential scattering of the model with varying DPb-Br, i.e. with displacements of the Br ions in direction parallel to the cell edge (the Pb … Pb vector, note that here the tilt angle φ is held fixed).With these displacements, there is negligible change in the relative intensity of the 211.Similarly, small and negligible changes to the 211 peak occur with Pb off centering, DPb.We can therefore conclude that the relative intensity of the 211 peak can be used to quantify the magnitude of the displacement of the Br ions in the direction perpendicular to the cell edge, and, therefore, we take this intensity change as a proxy for the φ angle, and the (180-φ) Pb-Br-Pb bending, changes.The measured strong photoinduced reduction of the 211 peak, therefore, indicates a strong reduction of φ, as discussed in the main text.The second largest feature observed in the experimental differential scattering profile occurs at q ~ 3.6 Å VP , which is approximately the position of the 222 peak.However, there are no distortions of the Pb-Br sublattice which selectively downsize the relative intensity of the 222 peak relative to the other high symmetry peak, e.g.200.This can be seen in Fig. S6, where the change in the intensity of the 222 peak follows closely that of the 200 for all possible distortions.This implies that the observed negative differential peak observed at q ~ 3.6 Å VP is not a result of a strong reduction of the 222, as this would contradict the slight enhancement observed for the 200.This feature is thus likely a result of a reduction of the other Bragg peaks close to the 222, or strong reduction of the diffuse scattering about the 222 peak.This feature remains, so far, uninterpreted.

Simulated differential scattering profiles
In Fig. S7 we plot the simulated differential scattering profiles for the model described above, using the q resolution of the UED experiments compared to the measured differential scattering.A photoexcited change in φ (S7a) gives a good agreement only at low q, as it correctly reproduces the strong reduction in the 211 peak, but not that at peaks falling near 3.6 and 5.6 Å VP .We next considered a photo-induced increase in ∆ TUVW= (S7b) in addition to a decrease in f, which would result from a Fröhlich-type coupling to the high energy optical phonons in leadbromides (~15-20 meV, see figure 3d of the main text).The simulated differential spectra are at complete odds with that measured, showing strong positive differential peaks in the 3.6 and 5.6 Å VP regions.Conversely, assuming a photoinduced decrease in nominal ∆ TUVW= (a regularization of the Pb-Br octahedra assuming finite distortions of them in the equilibrium structure), gives closer agreement (S7c).In this case, strong negative differential features appear close in q to the experimentally observed features.Similarly, if a nominal off-centering of the Pb ions in the equilibrium structure exists, (∆ TU ), a photoinduced reduction of this off-centering leads to strong negative differentials close to the experimentally measured features at ~ 3.6 and 5.6 Å VP (S7d).In the FAPbX3 species, the relatively large FA molecules will make the displacements of the halide ions sticking out in the A-cation cages probably correlated.As we discuss in the main text, we speculate that photoexcitation may also induce slight reorientations of the FA molecules, facilitating reduction of the PbBr6 octahedra tilts.Therefore, correlations in the photoexcited displacements of the Br ions within the A-cation cages (an effect that our model cannot capture), cannot be ruled out.More satisfactory fits to the differential spectra could result from considering such correlations, but would require a more detailed model for the equilibrium structure of the glassy FAPbBr3, which, presently, is not available.

Figure S7. Experimental differential scattering curve at 100K (red trace) vs simulated differential electron diffraction patterns (blue lines) using the model described above for FAPbBr3 cubic shaped nanocrystals, assuming photoexcitation induced a) decrease in tilt, b) decrease in tilt and increase in ∆ TUVW= , c) decrease in tilt and ∆ TUVW= , d) decrease in tilt and ∆ TU . Discussion
While it is not possible from the experiments to extract the precise photo-excited distortion of the Pb-Br sublattice beyond a reduction of tilting, we can use theory to elucidate possible mechanisms.We first consider displacements of the Pb ions from the center of the octahedra, ∆ TU .In Fig. S8 we plot the shift in the bandgap as a function of ∆ TU .Similar to the tilt dependence (Fig. 3b of the main text) the bandgap is minimized when the Pb ions are located at the center of the octahedra.This result can be explained using the same argumentation employed to explain the tilting dependence; that the magnitude of sp bonding and antibonding in the CBM and VBM will be maximized when the Pb ions are centered.In the idealized Pm-3m structure, there will be no coupling to phonons which drive ∆ TU within the structure, as  [ /∆ TU | ∆ ]^O _ = 0. Finite Pb-off-centering has been shown to be intrinsic in the equilibrium structure of FAPbBr3 (∆ TU > 0). 11Given this, there will be finite coupling to phonons which drive ∆ TU , and there will be a photoexcited decrease in the magnitude of ∆ TU .Analysis of the phonon density of states indicates that these phonons should occur over an energy range of ~9-13 meV (see Fig. 3d of the main text).

Figure S8. Plot of the bandgap shift as a function of Pb displacement (∆ TU ), the x-axis is in % of the nominal Pb-Br bond length
Next we consider the coupling of the high energy LO phonon, ~17 meV.The high energy optical phonons in the energy range 15-20 meV drive tetragonal-distortions of the octahedra, displacing Br ions in the lattice along directions parallel to the Pb-Br-Pb bonds (see Fig. 3d of the main text).In our structural model above, these phonons, including the high energy LO, will thus drive changes in ∆ TUVW= .In Fig. S9 we plot the bandgap as a function of the phonon normal coordinate of the high energy LO mode, QLO.A very weak dependence is observed for small QLO.In the ideal Pm3m structure with perfectly regular octahedra, the coupling to this mode is thus expected to be 0.However, if there are equilibrium distortions of the octahedra, the coupling become finite, and increases in magnitude as the magnitude of nominal ∆ TUVW= increases.Contrary to the coupling to tilting and Pb-off centering, in this case, EP-coupling to the LO will drive an increase in the octahedral distortions in the excited state.Our simulations assuming a photoexcited decrease in ∆ TUVW= which somewhat qualitatively matches some of the high q differential spectra (Fig. S7c), is at odds with the theory.Conversely, we have seen that a large photoexcited increase of ∆ TUVW= gives a differential spectrum that dramatically disagrees with the measured one (Fig. S7b).

Figure S9. Plot of the bandgap shift as a function of the shift of the normal coordinate of the highest energy LO phonon, QLO
A coupling of this mode to interband excitation of a single exciton of SLO ~0.05 has been extracted from single dot luminescence spectra on ~9 nm FAPbBr3 NCs, 12 meaning a shift in  #a = b2 #a = 0.25.From the LO phonon eigenvectors calculated for Pnma CsPbBr3, a shift For small deformations ( > 0) from the orthorhombic structure upon the photoexcitation of Nex excitons, we approximate the total energy as Here  _ is the total energy of the unexcited orthorhombic phase, the second term is the increase in formation energy of the distorted structure which is quadratic in  x y ,€ for small  (see Fig. S10b), and the third term is the decrease in energy of the  ƒ" excitations relative to the orthorhombic phase, where  ƒ" / is the derivative of the excitation's energy with respect to  (see Fig. S11).As in the main text, we assume here that energy of each exciton scales proportionally to the bandgap.NU appearing in eq.S3 is the number of APbX3 units over which the Nex excitations are localized, and we assume that the cubic distortion occurs uniformly over this volume.There can be additional terms, such as exciton binding energies, carrier-carrier interaction etc., however we assume here that these terms scale negligibly with , and do not write them out explicitly.We can then find the value for  which minimizes eq.S4,  ˆ, by solving  ‚<‚ / = 0, The electron phonon coupling strength resulting from octahedral tilting can then be calculated by its definition Both the magnitude of the cubic distortion ( ˆ) and the coupling  Š x y will depend on the number of unit cells over which the excitation is confined (both being proportional to  { VP ).
The polaron diameter, dp, resulting from coupling to a phonon of frequency w is given by 15 where  ƒ" is the effective mass of the excitation (i.e.electron, hole, or exciton effective mass).
Taking the CsPbBr3 exciton effective mass as 0.1me, 16 we can estimate the lower limit of the polaron diameter by computing  • for coupling to the ~17 meV high energy LO phonon, which gives ~14 nm.For CsPbBr3 NCs, this is larger than the NC, and we can assume a polaron volume equal to the volume of the NC.In Fig. S12b we plot the computed  Š x y for a single exciton on a CsPbBr3 NC.
In Fig. S11c,d we plot the shift of the VMB and CBM as a function of octahedral tilt.An increase in the VBM and decrease in the CBM with increasing tilt is apparent, and the eq.S6 can be used to compute the coupling of a bare charge, using  ' / ~ -447 meV for holes and  ƒ / ~ -480 meV for electrons assuming a fixed unit cell volume.These values suggest phonon-mediated attractive electron-electron and hole-hole interactions.We note that to arrive at the values of  ƒ / and  ' /, the absolute position of the bands for the independent DFT calculations are aligned to a reference, which always carrier the risk of a systematic error in the extracted position of the bands.We used here the lowest 16 non-bonding Pb d-bands, occurring ~17.3 eV below the VBM as the reference for each calculation.

Supplementary Note 5: Extracting Electron-Phonon Coupling Strengths from Measured Lattice Reorganization
Coupling to octahedral tilting: From the UED measurements in the main text, we observe a ~3% decrease in the 211 peak intensity in the FAPbBr3 NCs at a fluence of 0.6 mJ/cm 2 corresponding to ~40 excitons.This 3% decrease corresponds to a decrease in the PbBr6 octahedra tilt of ~1.0 degree (Fig. 2d) and a  ˆ( ƒ" = 40) ~ 0.07 (see Note 4).As  ˆ is linear in  ƒ" , we take  ˆ( ƒ" = 1) ~ 0.0017.We then can estimate the electron phonon coupling strength resulting from octahedral tilting for a single exciton as  Š x y = 7 ˆx y ,€ 9 E /2, the results are plotted in Fig. S12c (and in Fig. 3e of the main text).We note that the extracted coupling strengths depend upon the assumed polaron radius (through NU in eq.S3), and above we assume all excitons overlap over the entire NC based on the estimated polaron radius in Note 5.More generally, without this assumption we can write  ˆ( ƒ" = 1) =  ˆ( ƒ" ) where VNC is the volume of the NC and VP is the polaron volume, which accounts for the fraction of the NC volume contributing to the measured signal.As  x y ,€ ∝ b T is also proportional to the polaron volume, the coupling strengths then scales with the assumed polaron radius  Š x y ∝  T VP .In Fig. S13 we plot the extracted coupling to the 6 meV mode as a function of polaron radius.The assumption of VP = VNC gives an excellent match to previously measured coupling strengths of this mode in similarly sized NCs, where a coupling to a 5 meV mode of ~ 0.15 -0.35 was estimated for similarly sized NCs. 12 The extracted coupling strength become unreasonably large as the polaron size is reduced below that of the NC volume, even for large overlapping polarons (e.g.VP =1/3 VNC gives S = 1).This is supportive of the assumption that VP = VNC.

Additional Coupling
As discussed in Supplementary Note 3, in addition to a photoinduced reduction in tilt, there is likely renormalization of the Pb-Br bonds.Here we assume this renormalization stems from a photoinduced centering of nominally off-centered Pb ions wthin the Br octahedra.In Fig. S14a we plot the fluence dependence of change in the [311] peak extracted from the MeV-UED at a q~3.6 Å -1 , which is linear in fluence within the range measured.We extract a change of ~2% at a fluence of 0.6 mJ/cm 2 .In Fig. S14b we plot the relative increase in the 311 peak intensity as a function of Pb-shifts in the split-cubic structure, averaged over Pb-shifts in the [111] and [100] direction.The change in the 311 with Pb-shifts is non-linear for small equilibrium shifts, which complicates the extraction of  Š x y , as it will depend on the initial equilibrium shift.Previous reports have estimated an equilibrium shift of ~0.13 Å in FAPbBr3. 11This estimate is in the highly nonlinear portion of the scattering intensity scaling curve however (Fig. S14b), which is at odds with the linear scaling we observed in Fig. S14a.We therefore assume an initial Pb-shift of ~0.2 Å which is then within the region in which the scaling becomes reasonably linear.For extraction of the coupling we then take that 0.6 mJ/cm 2 photoexcitation causes on average a 0.06 Å reduction of the Pb atoms nominally shifted by 0.2 Å.Then, to estimate the coupling, we generate structures with Pb atoms within the Pnma structure shifted along the different [111] and [100] directions from the center of the octahedra, and calculate  x y () where  = 0 We show in the following that an entropic contribution to the free energy of glassy structure may also enhance coupling at elevated temperatures.At finite temperatures, the energy of the lattice (Eq.S4) should be replaced with the free-energy, where T is the temperature and () is the entropy.For small , which gives a correction in the denominator of the coupling strength (eq.S6) We refrain from attempting to determine an expression for (), and note that the stabilization of the average cubic phase at high temperatures in LHPs indicate that ()/ is finite and positive.From eq.S11 we can therefore conclude that the coupling strength will increase with increasing temperature as observed in our temperature dependent measurements (Fig. 4b) and will depend on the magnitude of ().A similar correction to the single mode model in the main text can be similarly done with free energy from which the same conclusions can be drawn.
Finally, enhancement of the coupling to the low energy optical modes may stem from their correlation to anharmonic coupling of the rotational modes of the FA ions, such that photoexcitation additionally causes a reorientation of the FA. 17,18pplementary Note 7: FLUPS Measurements on FAPbBr3 and CsPbBr3 NCs In Fig. S16a we plot the total integrated (over energy and time) PL intensity for CsPbBr3 and FAPbBr3 NCs as a function of pump power.While initially increasing with pump power, the total integrated PL saturates at low pump powers, indicative of a rapid decrease in the PL quantum yield (PLQY) with increasing exciton density NEX.We assume here a form Eq. S14 is plot in Fig. S16b.This form for the PLQY gives a value of ~1 for low excitation densities (NEX < 1) , consistent with the very high PLQY expected in this range, and ~1/ NEX for NEX >> 1, i.e. for large NEX we expect on average a single photon out, independent of NEX.This assumed form for the PLQY, we obtain a reasonable agreement to the total integrated PL as a function of pump power (dashed line in Fig. S16a).We then use eq.S14 to compute the relative contribution to the FLUPS signal as a function of NEX for a gaussian pump beam profile.In Fig. S16c we plot the relative contribution for 3 pump powers.The conclusion from these plots is that the dominant contribution to the FLUPS signal will be from weakly (small NEX) excited NCs at the periphery of the beam where the PLQY is high.
In Fig. S17 we plot the total PL as a function of time for varying pump powers (from 3 to 75 nJ) in FAPbBr3 and CsPbBr3 NCs.The average relaxation time constant from our UED measurements,  = , is ~35 ps.For both NCs, multiexciton-decay rates are reasonably fit with a fluence independent time constant of this same value.For FAPbBr3 NCs, a fast decay component with time constant ~10 ps is also observed for high pump intensities.We note that the maximum fluence used in the FLUPS measurements are ~5x more intense than in the UED measurements (~4.5 mJ/cm 2 peak fluence for the 75 nJ power), and speculate that this fast component may stem from very high excitation densities.

Figure S1 .
Figure S1.Normalized time resolved differential scattering of optically pumped FAPbBr3 NCs.Pump fluence and temperature of the measurement are shown on each map.

Figure S2 .
Figure S2.Plots of the weighted differential scattering (eq.S1) along with corresponding biexponential fits.Pump fluence and temperature of the measurement are shown on each plot.Error bars represent 1s uncertainty.

Figure S3 .
Figure S3.Plot of the magnitude of the differential scattering at 5ps as a function azimuthal angle (f in panel a), which indicates an isotropic response of the lattice.Data at 100 K with 0.8 mJ/cm 2 is shown here.

Figure
Figure S6.a) diagram of the split-cubic model; b) diagram of the split-cubic model including displacements of the Br ion parallel to the Pb…Pb vector (∆ TUVW= ) and Pb displacements (∆ TU ).c) plot of the simulated electron diffraction profile for the split cubic model with φ=0 and φ=14.6°, with ∆ TUVW= = ∆ TU =0.d) simulated diffraction profile for the model with varying ∆ TUVW= .e) simulated diffraction profile for the model with varying ∆ TU .Negative differential scattering at 3.6 Å VP

Figure S11 .
Figure S11.a) Calculated bandstructure of orthorhombic CsPbBr3.The bandstructure in regions defined by the black boxes are shown on the right for g varying from 0 to 1. b-d) Plot of the energy redshift for excitations in CsPbBr3 as a function of g, where g = 0 corresponds to the orthorhombic phase and g = 1 the cubic phase.

Figure S12 .
Figure S12.a) Computed shift of normal coordinates  x y ,€ between the orthorhombic and cubic phase of CsPbBr3.b) The calculated electron-phonon coupling strength  Š x y to octahedral tilting per exciton in CsPbBr3 NCs.c)  Š x y for FAPbBr3 NCs estimated from the magnitude of the lattice reorganization measured with time resolve electron diffraction.

Figure S13 .
Figure S13.Plot of the extracted coupling to the 6 meV mode as a result of octahedral tilting as a function of assumed polaron volume.Assuming a polaron volume equal to the NC volume gives provides an excellent match to experimentally measured couplings, while smaller polaron sizes yield unreasonably large coupling strengths.

Figure
Figure S16.a) Total measured PL intensity, integrated over time and energy, for FAPbBr3 NCs (blue circles) and CsPbBr3 NCs (open red circles).The dashed line shows the simulated intensity assuming a PLQY(NEX) in eq.S14 and shown in (b).c) Plot of the relative contribution to total FLUPS signal as a function of NEX for various pump intensities.For all pump powers, a majority of the FLUPS signal stems from the PL from weakly excited NCs in the periphery of the beam (where PLQY is high).

Figure S17 .S20Figure S18 .
Figure S17.Plot of the total PL as a function of time for varying pump powers (red to blue = 3 to 75 nJ) in FAPbBr3 (a and b) and CsPbBr3 NCs (c).For both NCs, multiexciton-decay rates are reasonably fit with a fluence independent time constant of ~35 ps.For FAPbBr3 NCs, a fast decay component with time constant ~10 ps is also observed for high pump intensities.

Figure S20 .
Figure S20.Emission and absorption spectra of nanocrystal samples.