Intensity / a.u. 2 theta / deg. MAPbI 3. 1:1 MaPbI 3-x. Cl x 3:1. Supplementary figures

Transcription

1 Intensity / a.u. Supplementary figures 110 MAPbI 3 1:1 MaPbI 3-x Cl x 3: theta / deg Supplementary Fig. 1 X-ray Diffraction (XRD) patterns of MAPbI3 and MAPbI 3-x Cl x spin-coated on a glass slide and annealed at 100 C. Symmetric -2 scans collected for films prepared by spin coating from different solutions of precursors: stoichiometric 1:1 mixture of MAI of PbCl 2 in DMF to obtain the pure iodine phase (black pattern); non stoichiometric 3:1 mixture of MAI and PbCl 2 to promote the formation of the I 1-x Cl x mixed phase (green pattern). For both films, all Bragg peaks were indexed as (hh0) Bragg reflections on a tetragonal I4/mcm cell with (with,, indicating a highly oriented growth, with films having the face of the cubic perovskite cell lattice parameter) parallel to the deposition plane

2 Supplementary Figure 2 Photoluminescence and UV-Vis absorption of the MAPbI 3-x Cl x film. UV-Vis absorption spectra are shown for room temperature (red line) and 170K (blue line), together with the corresponding photoluminescence spectra (dotted red line 300K, dotted blue line 170K).

3 Supplementary Figure 3 Photoluminescence spectra corrected for self-absorption. Upper panel: UV-Vis absorption (dashed line) and steady-state photoluminescence spectra for the MAPbI 3 film kept at 170K; the continuous ( blue line is the photoluminescence spectrum multiplied by ), where is the effective optical density of the film for photon energy, calculated considering that photoluminescence is collected in transmission geometry and therefore the emitted light has to cross the entire film before being measured. Spontaneous emission was excited by a 100 nm, a small fraction of the film thickness, 800 nm. Lower panel: same measurements as in upper panel, only temperature is varied to 300K.

4 Supplementary Figure 4 Simulation of the UV-Vis absorption spectrum of Perovskite films: sensitivity on the exciton binding energy. The absorption spectrum for the MAPbI 3 film (dotted black line) is compared with three simulations based on Supplementary eq. 9 with three different values for the exciton binding energy, namely 20 mev (green line), 25 mev (red line) and 30 mev (blue line). The discrepancy between the curves demonstrates the sensitivity of the fitting on the exciton binding energy.

5 Supplementary Figure 5 Transient photoluminescence spectra. Upper panel: transient photoluminescence spectra for the MAPbI 3 integrated in a time window corresponding to the instrument resolution (60 ps); injected carrier density, estimated from laser pulse fluence and film absorbance, are reported in the legend. Lower panel: same measurements as in top panel, but on the MAPbI 3-x Cl x film.

6 Supplementary Figure 6 Transient photoluminescence spectroscopy in trihalide perovskites: failure of the equilibrium model based on Saha equation. Photoluminescence emission intensity estimated at t=0 as a function of injected electron-hole density (lower axis) and laser pulse fluence (upper axis). The quadratic dependence is shown by the black dotted lines as a guide for eyes. Dotted lines: intensities calculated according to the Saha equation. Simulations consider the exponential spatial profile of the electron-hole density created by laser pulses. is calculated by summing the spontaneous emission intensity emitted by the photoexcitations along the light absorption path. Saha equation predicts that saturation occurs at an injection density one order of magnitude smaller than what observed experimentally.

7 Supplementary Figure 7 Quasi-chemical potential in an ionized plasma and in a gas of bound and unbound excitons in thermal equilibrium. The chemical potential is calculated for an electron-hole plasma (dashed blue line) and for a mixed gas of bound excitons and unbound electron-holes in thermodynamic equilibrium, as predicted by Saha equation (dotted black line) or by assuming a Fermi-Dirac distribution of free carriers and a Bose-Einstein statistics for excitons (solid green line). Optical gain occurs for photon energies below the chemical potential,. Stimulated emission at ω= is not allowed for if the ratio between the populations of bound and unbound excitons are in thermal equilibrium.

8 Supplementary Note 1 of bound and unbound excitons in thermal equilibrium. In thermal equilibrium, the balance between exciton formation and ionization (e+h X) is dictated by the law: where is the equilibrium population constant. This latter depends on the exciton binding energy and the thermal wavelengths of electrons, holes and excitons. We took, which give a reduced exciton mass in reasonable agreement with theory and experiments 1 3. Under optical excitation, an equal number of holes and electrons are excited,, and. At low carrier densities, exciton dissociation is more efficient than exciton formation. According to Supplementary equations 1-2 the majority of carriers is unbound, and the population of bound states is negligible. In the opposite limit, the equation holds owing to the fact that the exciton formation rate increases as. Hence, the Saha equation predicts that the electron-hole gas undergoes a transition from a conducting plasma to an insulating excitonic phase for increasing the electronhole density. To the best of our knowledge, this transition has never been observed in photoexcited semiconductors. In the case of trihalide perovskites, from the estimated exciton binding energy and reduced exciton mass and the calculated thermal wavelengths, we found 2.8 x cm -3, well below the density threshold for amplified spontaneous emission, and easily accessible experimentally. We discuss the impact of the equilibrium law on the photoluminescence when the injected electron-hole density is tuned across the density. The spontaneous light emission intensities from free carriers and excitons can be expressed in the following forms: where is the bimolecular radiative recombination rate of unbound electrons and holes, is the monomolecular radiative rate of the exciton population. The equilibrium condition has a simple but important consequence: the contribution of excitons and free carriers to the photoluminescence is independent of the photoexcitation densities: their relative weight to the photoluminescence intensity remains constant and does not affect the overall dependence on the excitation rate. This is indeed consistent with the almost fluence-independent photoluminescence spectrum we observed in the experiments. At low carrier densities, and is proportional to the square of the injected carrier density,. Both the exciton and free carrier photoluminescence scale quadratically with. even though the radiative decay of excitons is a monomolecular process. In the opposite limit, and ; the linear scaling with the injected carrier densities is satisfied, again, both by bound and unbound electron-hole pairs. The expected theoretical dependence of on the pulse fluence is reported in Supplementary Fig. 6, according

9 to Supplementary equations 1-3. was calculated taking into account the spatial excitation profile caused by the strong absorption coefficient at the pump wavelength. The transition from an ionized plasma to a gas of bound exciton manifests as a change from a quadratic to a linear behaviour of as a function of As expected, this deviation occurs at an injection density one order of magnitude smaller than what observed experimentally. This discrepancy implies a considerable overestimation of the excitonic population for cm -3 by the Saha equation. Supplementary Note 2 Chemical potential in a gas of bound and unbound excitons. The chemical potential of the electron and hole gas with density was calculated by solving the equations: ( ) Where is the density of electron (hole) states. The factor of two takes into account the spin degeneracy of the conduction and valence bands in perovskite films 3. In Supplementary equation 4, are calculated with respect to the band extremes. In the case of a gas of unbound electrons and holes in thermal equilibrium with excitons, the density of free carriers was determined from the Saha equations (Supplementary eqs. 1-2), and then were calculated by solving the equations: ( ) The Saha equation is valid in the dilute density regime as it assumes a Boltzmann statistics for electrons, holes and excitons. We also calculated according to a Fermi-Dirac distribution for freecarriers and a Bose-Einstein statistics for excitons: [ ] [ ] where the factor 4 stems from the degeneracy of the two-particle states, is the Bose function, and is the density of exciton states. Supplementary Fig. 7 shows that the two simulations provide undistinguishable results in the regime of interest. Optical gain occurs for photon energies below the chemical potential, ω. If the ratio between the populations of bound and

10 unbound excitons were in thermal equilibrium, stimulated emission at allowed for owing to the fact that at this density. would not be Supplementary Methods. Excitonic absorption coefficient in the Elliott s theory. To go beyond the approximation of parabolic bands, we introduced the joint valence-band energy-momentum dispersion Where is the reduced electron-hole mass, Supplementary equation 7 gives the joint density of states: Supplementary equation 8 recovers the usual density of state of parabolic bands for expression used to fit the experimental absorption curves was:. The [ ] For when the excitonic enhancement becomes negligible, the contribution of continuum states to the absorption coefficient correctly scales as the density of states expressed by Supplementary equation 8, as expected for bare band-to-band transitions. Experimental curves were fitted with. The secant hyperbolic function of width accounts for line broadening. To check the sensitivity of the absorption coefficient on the binding energy, Supplementary Fig.1 reports at T=300 K for different values of, but keeping constant both line broadening and band nonparabolicity. Rate equations for steady-state photoluminescence. The rate equations for the electron, hole and trapped carrier densities in stationary conditions to simulate steady state photoluminescence read:

11 the trapping rate for electrons, the Auger recombination coefficient, the carrier injection rate provided by laser excitation, the bimolecular recombination rate between trapped electrons and holes, the density of available traps and therefore the maximum density of trapped electrons. The main assumption is that only electrons are trapped, holes are never captured. From Figure 2, was approximated by the saturation formula: where and The power law is quite robust against the choice of and. We fixed the following relations between the bimolecular recombination coefficient and the coefficients and :, ( = 2.6 x cm 3 s -1 ); we set in such a way to match the trapping rate of electrons with the low-density decay rate of the photoluminescence. The density of traps could vary in a wide range, approximately up to, without altering the power dependence of the PL signal on the laser intensity. Saturation of the trapped electron population sets in when the radiative bimolecular recombination rate overcomes the trapping rate. Supplementary References 1. Hirasawa, M., Ishihara, T., Goto, T., Uchida, K. & Miura, N. Magnetoabsorption of the lowest exciton in perovskite-type compound (CH 3 NH 3 )PbI 3. Phys. B 201, (1994). 2. Tanaka, K. et al. Comparative study on the excitons in lead-halide-based perovskite-type crystals CH 3 NH 3 PbBr 3 CH 3 NH 3 PbI 3. Solid State Commun. 127, (2003). 3. Giorgi, G., Fujisawa, J., Segawa, H. & Yamashita, K. Small Photocarrier Effective Masses Featuring Ambipolar Transport in Methylammonium Lead Iodide Perovskite: A Density Functional Analysis. J. Phys. Chem. Lett. 4, (2013).