Enhanced screening and spectral diversity in many-body elastic scattering of excitons in two-dimensional hybrid metal-halide perovskites

In two-dimensional hybrid organic-inorganic metal-halide perovskites, the intrinsic optical lineshape reflects multiple excitons with distinct binding energies, each dressed differently by the hybrid lattice. Given this complexity, a fundamentally far-reaching issue is how Coulomb-mediated many-body interactions --- elastic scattering such as excitation-induced dephasing, inelastic exciton bimolecular scattering, and multi-exciton binding --- depend upon the specific exciton-lattice coupling. We report the intrinsic and density-dependent exciton pure dephasing rates and their dependence on temperature by means of a coherent nonlinear spectroscopy. We find exceptionally strong screening effects on multi-exciton scattering relative to other two-dimensional single-atomic-layer semiconductors. Importantly, the exciton-density dependence of the dephasing rates is markedly different for distinct excitons. These findings establish the consequences of particular lattice dressing on exciton many-body quantum dynamics, which critically define fundamental optical properties that underpin photonics and quantum optoelectronics in relevant exciton density regimes.

In two-dimensional hybrid organic-inorganic metal-halide perovskites, the intrinsic optical lineshape reflects multiple excitons with distinct binding energies, 1,2 each dressed differently by the hybrid lattice. 3 Given this complexity, a fundamentally far-reaching issue is how Coulomb-mediated many-body interactions -elastic scattering such as excitation-induced dephasing, 4 inelastic exciton bimolecular scattering, 5 and multi-exciton binding 6,7 -depend upon the specific exciton-lattice coupling. We report the intrinsic and density-dependent exciton pure dephasing rates and their dependence on temperature by means of a coherent nonlinear spectroscopy. We find exceptionally strong screening effects on multi-exciton scattering relative to other two-dimensional single-atomic-layer semiconductors. Importantly, the exciton-density dependence of the dephasing rates is markedly different for distinct excitons. These findings establish the consequences of particular lattice dressing on exciton many-body quantum dynamics, which critically define fundamental optical properties that underpin photonics and quantum optoelectronics in relevant exciton density regimes.
Spectral transition linewidths provide pertinent insights into the system-bath interactions in materials because they depend on optical dephasing dynamics -the processes by which the coherence that the driving electromagnetic wave imparts on the optical response dissipates due to scattering processes with lattice phonons, other excitations, and defects.
Dephasing rates thus are very sensitive probes of the consequences of lattice dressing effects on excitons. Nevertheless, these are challenging to extract directly from linear optical probes such as absorption or photoluminescence spectroscopy given that the experimental linewidths typically arise from two distinct but co-existing contributions: homogenous and inhomogenous broadening (see Fig. 1a). While the former is due to dephasing and is governed by the intrinsic finite lifetime of excited states and by dynamic disorder, the latter is caused by a statistical distribution of the transition energy due to static disorder, defects or grain boundaries. The exciton homogeneous linewidth 2γ (full width at half maximum) is limited by the exciton lifetime (Γ −1 ) and the dephasing rate mediated by exciton-exciton elastic scatterting (excitation-induced dephasing, γ EID / ) and phonon scattering (γ ph / ): An accurate estimate of γ is thus crucial to quantify the magnitude of the inter-exciton and 10 -1 10 0 Absorbance (OD) 10 S1 for the temperature-dependent absorption spectra). We quantify γ as a function of exciton density and temperature and find that these different excitons display peculiar many-body phenomena.
Four-wave-mixing spectroscopies measure coherent emission due to a third-order polarization induced in matter by a sequence of phase-locked femtosecond pulses, and its dissipation reports directly on dephasing processes. In a two-dimensional coherent excitation geometry, one spectrally resolves this signal, and a 2D spectral correlation map of excita- while the diagonal one reads as Here erfc corresponds to the complementary error function, ω ad and ω d are the anti-diagonal and diagonal angular frequencies, respectively, and δω characterizes the inhomogeneous distribution. The sum in equation 3 runs over the relative amplitudes α j and the central diagonal energies ω j of optical transitions of excitons A, A , B and B*.
The laser spectrum used in this experiment, which covers all excitonic absorption features, is displayed in Fig. 1b indicating that all the transitions share a common ground state. An intense cross peak at an emission energy of 2.305 eV is observed for exciton A despite the presence of a much weaker diagonal feature at lower energy than that dominant peak. Given the opposite phase of this feature (the real part of the spectra shown in Supplemental Information Fig. S3) when compared to the corresponding diagonal peak, we attribute it to an excited-state absorption from a singly bound A exciton to a bound AA biexciton. 7 The diagonal and anti-diagonal cuts at the energies of excitons A, A and B, along with the best fits to equations 3 and 2, are plotted in Fig. 2a, b and c respectively. The only fit parameters are the amplitudes α j , homogeneous dephasing width γ and inhomogeneous width δω of each optical transition.
To assess the contribution of many-body interactions on the dephasing dynamics of the different excitons, we acquired 2D coherent excitation spectra for a wide range of excitation fluences and sample temperatures (see Fig. S4 and S5 of Supplemental Information for the raw data used here). The monotonic rise of γ with the excitation fluence at 5 K is shown in Fig. 3. Such a dependence on exciton density n is a consequence of broadening induced by exciton-exciton elastic scattering mediated by long-range Coulomb interactions: Here, γ 0 is the density-independent (intrinsic) dephasing rate and ∆ is the exciton-exciton interaction parameter. Excitons in 2D metal-halide perovskites are confined to one of the test the robustness of the fits and to estimate the uncertainties on the extracted linewidths, we repeated the fitting procedure numerous times while adding white noise (5% of the cut's maximum peak to peak) to the data.
inorganic quantum wells and are electronically isolated from the others due to the large inter-layer distance 3 (∼ 8Å ) imposed between them by the long organic cations. However, the sample itself, 40-nm thick, is composed of tens of these quantum wells, leading to a highly anisotropic exciton-exciton interaction. To allow for some form of comparison with other 2D semiconductors and quasi-two-dimensional quantum wells of similar thicknesses, we report the exciton-exciton interaction parameter, ∆ in units of energy per area. The associated fits and the fit parameters are displayed in Fig. 3a, b and c. While γ 0 is approximately revealed ∆ = 2.7 × 10 −12 meV cm 2 and 4 × 10 −13 meV cm 2 , respectively, three and two orders of magnitude higher than the value obtained here for (PEA) 2 PbI 4 . This and the linearity of the dephasing rates over a wide range of excitation densities 12 highlights the substantial screening of the exciton-exciton interactions in these 2D perovskites. This is especially surprising given the high biexciton binding energy, 7 another characteristic that (PEA) 2 PbI 4 shares with TMDC monolayers. 13,14 We will return to these differences in ∆ below.
To further highlight differences in multi-exciton elastic scattering behavior, we explore the dependence of the laser polarization state on γ, reported in Table I Table I further points to differences in scattering behavior of distinct excitons.
We now turn to the temperature dependence of γ to investigate exciton-phonon interactions. The measured γ for excitons A, A and B as a function of temperature are presented in Fig. 3d, e and f, respectively. We fit this data with which assumes that line broadening arises from scattering of excitons with a single thermally populated phonon mode of effective energy E LO with an effective interaction parameter α LO .
The parameters representing the best fit for each exciton are shown in Fig. 3d and f. This We consider that polaronic effects rationalize the two peculiar observations of this work in the context of many-body interactions in 2D perovskites -the remarkably low values of ∆, and non-trivial differences of this parameter and ∆ ph for diverse exciton resonances.
Remarkably low inter-exciton scattering rates along with relatively weak exciton-phonon interactions may be attributed to a polaronic protection mechanism, 19,20 where Coulomb interactions that are at the heart of both of these scattering events are effectively screened by the dynamic lattice motion. Such a mechanism has been invoked in the case of bulk 3D perovskites to explain slow cooling of hot carriers 21 and long carrier lifetimes, 22,23 and analogous comparisons can be drawn in their 2D counterparts. Excitons reorganize the lattice along well-defined configurational coordinates, 3 which dresses the exciton with a phononcloud. Although we currently do not have a reliable estimate of the spatial extent of this deformation, i.e., the polaron radius, based on the polaron coupling constants in lead-halide perovskites, 20 we may hypothesize that the radius is much larger than the exciton radius.
Such polaronic contributions effectively screen exciton-exciton interactions. At higher densities, we expect a high probability of multiple excitons within the polaron radius, in which case biexcitons may be populated given their high binding energy. 7 Lastly, we note that the zero-density (γ 0 ) and the zero-temperature (γ T =0 ) dephasing widths for all the excitonic transitions are between 2-3 meV, which corresponds to a pure dephasing time T * 2 = /γ 0,T =0 ≈ 500 fs. Time-resolved photoluminescence measurements performed at 5 K have revealed a lower limit for the exciton lifetime to be around 100 ps, 24 suggesting a radiative width γ rad ∼ 0.04 meV, which is much lower than the measured lowerlimit γ 0,T =0 . This clearly demonstrates the presence of an additional dephasing mechanism in addition to inter-exciton and phonon scattering, possibly due to the defective nature of the polycrystalline film, the presence of other degenerate dark states which excitons scatter in and out of, or via low-energy acoustic phonons unresolved in the current experiment.
We note that similar subpicosecond dephasing times were measured at low temperature for photocarriers in CH 3 NH 3 PbI 3 , which were noticed to be a factor of ∼ 3 times longer than in bulk GaAs. 25 This was ascribed to weaker Coulomb interactions in that perovskite, limiting the role of excitation-induced dephasing effects compared to III-V semiconductors. In ref. 25, it was speculated that dynamic large polarons account for the relatively long photocarrier dephasing time. We point out that these relatively slow dephasing rates are still much too fast for quantum optoelectronics applications such as single-photon emitters, 26 which require coherence times approaching the radiative lifetime.
In a general sense, multi-exciton interactions are determined by an interplay of Coulomb forces, and in ionic crystalline systems, the role of the lattice in mediating them is of fundamental importance. We have focused on the comparison of elastic scattering processes of spectrally distinct excitons within the excitation fine structure, manifested via the dephasing rate. In a previous publication, we have reported high biexciton binding energies in this material, 7 and importantly, that biexcitons display distinct spectral structure. We highlight that in ref. 7, the biexciton binding energy of excitons A and B appears to be different, and exciton A displays clear evidence of repulsive interactions in the two-quantum, two-dimensional spectral correlation map. Such interactions might give rise to inelastic scattering of exciton A, perhaps related to Auger recombination. 5 These overlapping dynamics would be deterministic in biexciton lasers, 27 for example, if these devices were to be rigorously implemented. The extent to which the spectral scattering rates depend on spectral structure might also determine the dynamics of exciton polaritons in semiconductor microcavities, in which quantum fluids are formed by polariton-polariton inelastic scattering. 28 It has been demonstrated that polariton-polariton interactions in 2D lead-halide perovskites are strong, 29 such that the ∼ 0.5-ps intrinsic dephasing times reported here are long compared to Rabi oscillation periods given 150-meV Rabi splittings. 30 In the search for room-temperature polartion condensates, it is important to know how polaronic effects control exciton many-body dephasing dynamics, and how hybrid perovskites differ in this respect from other two-dimensional candidate semiconductor systems such as monolayer TMDCs. 4,11 If polaronic effects mitigate excitation induced dephasing in 2D hybrid perovskites, then the quantum dynamics in a vast range of structurally diverse derivatives merit profound experimental and theoretical investigation.

Sample preparation
Thin films of (PEA) 2 PbI 4 (thickness of 40 nm) were prepared on sapphire substrates (op-  FIG. S1. Temperature dependent absorption spectra on a logarithmic scale.

ESTIMATION OF THE INDUCED EXCITATION DENSITY
The presence of spectral features in the absorption spectra as well as the pulses' nonuniform spectra must be accounted for when calculating the excitation density induced by one pulse. We do so by calculating the fluence to surface excitation density factor Θ as where A(E) is the absorption spectrum, P (E) is the pulse spectrum normalized so that its integral yields unity and E the energy. To obtain the surfacic excitation density, we then multiply the pulse fluence by Θ. The volumic excitation density is then obtained by dividing this number by the thickness of the material, namely 40 nm. We obtained a value of 8.75×10 20 excitons per J for Θ.  FIG. S5. Norm of temperature dependent photon-echo spectra at 5 K. Each spectrum was normalized by the maximal value of its norm. The corresponding absorption spectrum is plotted above each photon-echo spectra.