Ultrafast terahertz snapshots of excitonic Rydberg states and electronic coherence in an organometal halide perovskite

How photoexcitations evolve into Coulomb-bound electron and hole pairs, called excitons, and unbound charge carriers is a key cross-cutting issue in photovoltaics and optoelectronics. Until now, the initial quantum dynamics following photoexcitation remains elusive in the hybrid perovskite system. Here we reveal excitonic Rydberg states with distinct formation pathways by observing the multiple resonant, internal quantum transitions using ultrafast terahertz quasi-particle transport. Nonequilibrium emergent states evolve with a complex co-existence of excitons, carriers and phonons, where a delayed buildup of excitons under on- and off-resonant pumping conditions allows us to distinguish between the loss of electronic coherence and hot state cooling processes. The nearly ∼1 ps dephasing time, efficient electron scattering with discrete terahertz phonons and intermediate binding energy of ∼13.5 meV in perovskites are distinct from conventional photovoltaic semiconductors. In addition to providing implications for coherent energy conversion, these are potentially relevant to the development of light-harvesting and electron-transport devices.

A X 1s→3p (green dots), and unbound carriers A eh DS (blue dots) are presented as ω 2 p . This behavior is fully consistent with the perovskite phase transition and it is highly unlikely to assign these features to PMMA.
In the Supplementary Information, we present more details on experimental techniques, samples, extra results, data analysis and theoretical modeling.

Supplementary Note 1: Introduction
Organometal halide perovskite materials have recently emerged as one of the most exciting photovoltaic and opto-electronic materials due in part to their unique properties such as strong visible-light absorption, high charge mobility, long charge-carrier lifetime and diffusion lengths.
The determination of exciton binding energy and conversion pathways remains challenging in the hybrid organic-inorganic perovskites. This key missing information serves as a foundation to design efficient photoconversion schemes and device architectures based on these materials. Since the exciton formation also intimately relates to the loss of electronic coherence and conversion between Coulomb-bound and unbound e-h pairs, such investigation in organometal halide perovskite provides rare opportunities of learning more about these fundamental quantum processes. The lack of quantitative, ultrafast quasi-particle spectroscopy tools and the inability to elucidate initial quantum dynamics in the hybrid perovskites seriously limit both the thorough understanding of their photoconversion mechanism, and our perspectives of developing coherent quantum devices.
Here, we present a clear-cut picture, with an unprecedented level of details, of the THz quantum and thermal transport in perovskite materials. We reveal the ps coherent dynamics, and PbI 2 (9.6 mg, 0.02 mmol) in γ-butyrolactone (4 mL) was injected into toluene (15 mL) while stirring, and allowed to stand for 2 h at room temperature. The product was isolated by centrifugation (5 min at 4500 rpm) and washing with toluene (5 mL).
We study a free-standing 550 µm thick perovskite-poly(methylmethacrylate) thin film (CH 3 NH 3 PbI 3 /PMMA) that was made of embedding µm size CH 3 NH 3 PbI 3 crystals (Fig. 1b in the main text) in PMMA matrix. CH 3 NH 3 PbI 3 (6 mg, 0.01 mmol) was dispersed in a solution of PMMA (0.13 g, 0.8 µmol) in toluene (3 mL), while sonicating and agitating until the mixture became homogeneous. A homogeneous solution of CH 3 NH 3 PbI 3 and PMMA in toluene was prepared, cast into a mold and allowed to dry to optical quality films under ambient conditions, as shown in SEM image of the cross-section of CH 3 NH 3 PbI 3 /PMMA film. (Fig. 1c).
Room temperature X-ray diffraction (XRD) measurements of CH 3 NH 3 PbI 3 /PMMA film and pure CH 3 NH 3 PbI 3 powder corroborate the inclusion of the perovskites into PMMA matrix, as shown in Fig. 1d of the main text.
To fully characterize the low temperature phase transition behavior in the samples, we  Figure 2) and X-ray diffraction (XRD) measurements of CH 3 NH 3 PbI 3 sample down to 6 K (Supplementary Figure 3). We paid special attentions for the behaviors across the tetragonal-to-orthorhombic structural phase transition at T S =160 K. The temperature dependent XRD measurements were performed using the CH 3 NH 3 PbI 3 powder (particles in µm size) at temperatures ranging from 6 K to room temperature (295 K). The powder was mixed in the copper sample holder with GE varnish (commercial product) using toluene as a solvent and let dry for 24 hours at ambient conditions; then the solidified material was polished to create a flat surface suitable for the powder XRD data acquisition. The data were collected on a Rigaku Outside this transition region, both absorption and PL data exhibit a simpler, redshift of band edge by decreasing the temperature, i.e., the 'Varshni' trend in lead composite semiconductors.
The spectra of the 790 nm (pump #1) and 399 nm (pump #2) pump beams are also plotted in Fig. 1e in order to compare them with the absorption spectra. This way, fundamentally different initial excitation conditions can be selectively generated by tuning the pump photon energy, e.g., "cold" excitons and hot e-h plasma can be generated via resonant excitation close to 1s-exciton or off-resonant into interband continuum. These enable the unique study of the distinct initial conditions and their ultrafast evolutions, as shown in Figs. 3b-3c in the main text.
Our samples containing individual single perovskite crystals free from interactions are much more preferable to extract the true fundamental physics of the general perovskite materials, e.g., ultrafast exciton formation pathways and associated coherent transient effects. Particularly, we aim to underpin the similarities and differences of the perovskite materials compared to conventional optoelectronic semiconductor materials such as GaAs, which is also in a single crystal form, in order to further establish the new organic-inorganic perovskite systems as new revolutionary materials for light-harvesting and electron-transporting applications. In contrast to our samples, it is well established that electronic coherence and charge transport of thin film samples are dominated by the microstructural features such as grains and grain boundaries ( 10-1000 nm), which make it extremely difficult to study the universal and intrinsic physics of the materials by using such samples. Nevertheless, we think that the sample comparison should be performed in the future work.

Supplementary Note 3: Experimental details
We perform optical-pump and THz-probe spectroscopy, which is driven by a 1 kHz Ti:Sapphire where n 0 is the averaged static refractive index obtained from static measurement including contributions from both perovskite crystals and PMMA matrix based on their ratio, ∆n is the pump-induced refractive index change on the sample surface, d is the pump effective penetration depth obtained from absorption measurement by considering the sample as a homogeneous medium, and x is the distance that the light travels in the sample with x=0 at the entrance surface and x=L at the exit surface. L is the sample thickness. The dielectric function extracted this way is actually the averaged effective dielectric function consisting of contributions from both perovskite crystals and PMMA matrix, e.g., ε effective (ω). Then, effective medium approximation (EMA) is applied in order to calculate the perovskite contribution ε perovskite (ω) from the averaged effective dielectric function. To do so, the space filling ratio (FR) of the sample ∼1.2%, calculated from the SEM images, is required, which leads to a simple relation of ε effective (ω) ≈ FR· ε perovskite (ω) for FR much smaller than 1 as shown in ref 31 . For simplicity, we simply write ε perovskite (ω) as ε(ω) throughout the paper. Therefore, the static complex dielectric function ε(ω) and its pump-induced change ∆ ε(ω, ∆t) = ε excited (ω, ∆t)− ε(ω) are numerically retrieved. The Our results reproduce the main features that are fully consistent with the prior studies.
For example, our temperature-dependent transient THz spectra of ∆σ 1 (ω) and ∆ε 1 (ω) after 790 nm excitation and at pump-probe delay ∆t=60 ps (Figs. 2b and 2c) clearly reveal four phonon bleaching modes B j (j=1-4) whose frequency and temperature dependence are consistent with prior observations 24,25 . Most intriguingly, two photoinduced absorption oscillators centered at ∼10.1 meV and 12.1 meV can be attributed to intra-excitonic transitions of 1s→2p and 1s→3p of the perovskite Rydberg states. A complete temperature-dependent THz conductivities ∆σ 1 (ω) from 8 K to 295 K, as shown in Fig. 2d, clearly show that the two internal excitonic transitions exhibit maximum amplitude at 8 K, which gradually decrease and finally vanish by raising the lattice temperature above the structural phase transition.
As shown in Fig. 4a (main text), the time evolution of the photo-generated exciton density faithfully follows THz time scan (black line) with 50 fs resolution, which display a two-step formation pathways with characteristic times much slower than pump pulse duration of 50 fs.
Please note that it has been well-established that time resolution taken at such a condition is limited by gate pulse duration of 50 fs instead of THz probe pulse. These formation times are distinctly different from other conventional photovoltaic materials.

Supplementary Note 4: Theory and fitting results
The simultaneously-obtained, frequency-dependent complex conductivity on a sub-ps time scale allows us to quantitatively monitor the dynamic evolution of excitons and unbound e-h plasma, which have distinctly different spectral features in pump-induced changes ∆σ 1 (ω) and ∆ε 1 (ω).
We construct a THz line-shape model consisting of three components: the THz dielectric function of resonant excitonic absorptions (1st and 2nd terms), resonant phonon bleaching (3rd term) plus a Drude-Smith (DS) component of unbound e-h plasma (4th term): The first two terms accounts exactly the internal quantum transition of the excitonic Rydberg states 1s→np (n=2,3) These account for the correlated, dissipative and inductive features, exclusively below T S , from the internal, resonant quantum transitions between the Rydberg states.
where f is the oscillator strength, ω 2 p is the plasma frequency N X e 2 /(ε 0 µ), N X is the exciton density, e, µ, and ε 0 are the electron charge, exciton effective mass, and vacuum permittivity, respectively. ω 1s→np is the excitonic 1s → np transition resonant frequency and Γ is the broadening. Therefore, the ω 2 p measures the population difference between the two Rydberg states involved in the transitions (Fig. 1a) Next, the third term in Eq. 1 describes the THz dielectric function from photoinduced phonon bleaching modes of the CH 3 NH 3 PbI 3 crystals (B j , j=1-4) and PMMA matrix (P k , k=1-2), which is given by where ∆F phonon j,k = f j,k · ( ω 2 p ) j,k , similar to parameter definition of Eq. 2, and additionally, ω j,k and Γ j,k are the phonon resonant frequencies and broadenings, respectively. Note besides the phonon modes in the sample, marked as B j in Fig. 2d  , ω j,k , and Γ j,k are varied to match the experimentally obtained phonon strengths, resonant frequencies, and bandwidths for B j and P k , respectively. Note the high frequency mode at 26.7 meV (6.5 THz) only contributes as a "featureless" background in ∆ε 1 (ω) since the measured spectra region is below 3.5 THz (14.5 meV) (Figs. 2b and 2c).
The last term in Eq. 1 describes the non-resonant component from the Drude-Smith term where the plasma frequency ω 2 p = N eh e 2 /(ε 0 µ e ), N eh is unbound free charge carrier density.
And e, ε 0 , and ε ∞ are the electron charge, vacuum and background permittivity. γ is the electron broadening. The electron effective mass µ e =0.19m e is taken from Ref 36 . When the backscattering coefficient c 1 =0, the standard Drude model under the free electron approximation is recovered. When c 1 =-1, the DC conductivity is vanishing which manifests as backscattering of carriers. The photoinduced conductivity ∆σ 1 (ω) is suppressed (Fig. 2b) and ∆ε 1 (ω) increases rapidly (Fig. 2c) as The exciton and free charge carrier densities in our sample are calculated by fitting the integrated spectral weight of extracted THz conductivities and dielectric functions. We understand the accurate estimation of densities in such a highly inhomogeneous medium with randomly dispersed perovskite microcrystals is challenging. In the following we will compare the calculation of carrier densities from 2 different methods, i.e., from (1) THz conductivity and dielectric function and (2) absorbed pump photon density. Theoretically, the two should be very similar. The results show that densities are very similar for 790 nm excitation, and have a little difference for 399 nm excitation, the possible reasons of which will be discussed later.
Additionally, we also notice big scattering of visible light from perovskite crystals (Fig. 1e) in the absorption measurement, as confirmed by another perovskite study 29 , which justifies the relatively high pump fluences used against the moderate excited carrier densities calculated.
According Ref 29 , and using the optical density of an individual crystal from this paper as a reference, we get for our sample the perovskite crystal coverage ratio ≈1, scattering coefficient ≈0.945, and thickness coefficient ≈17. 35. This implies our sample is almost fully covered with perovskite crystals. Although the filling ratio is only ∼1.2%, the perovskite/PMMA film is pretty thick ∼550 µm, so it is not surprising that they can fully cover the film where measurement is taken.
This also suggests the effective thickness of the perovskite crystals to be 6.6 µm. In addition, which gives f 1s→2p = 0.416 and f 1s→3p = 0.079. Since excitons are composite bosons, with repulsive interaction to the second order, formed in a fermion many-body systems, one may not simply expect a pure Bose-Einstein or rigorous Fermi-Dirac distribution functions of hot excitons. Assuming that quasi-equilibrium has been reached at early stage and the Fermi-Dirac distribution, we further extract the effective temperature T * evolution of the excitons using the where the density of states of the ns/p excitons in the three-dimensional space is given by Given the experimentally-extracted ratio ∆N 1s,2p /∆N 1s,3p (inset, Fig. 4d), we calculate the effective temperature T * using the above relations. Fig. 4d in the main text shows that excitons stimulated by the 399 nm pulses are initially at quite high temperature, and gradually cool down on 10s of ps time scale. Meanwhile, the excitons generated by the 790 nm pulses are already very cool, close to the lattice temperature at the early stage. Finally, Fig. 4c shows that the photoexcitation quasi-instantaneously convert into mobile carriers, despite the absence of longer range transport hindered by disorder and/or crystal boundary as seen in the suppressed ∆σ 1 (ω) as ω → 0. 4a and 4e to the loss of exciton coherence, τ fast ∼1.0±0.03 ps, and the phonon-assisted processes that involve the low energy discrete phonon states in our system, τ slow ∼11.2±1.06 ps at 8 K, which lead to a redistribution of exciton states with different CM momenta along the 1s parabola.