Exciton-acoustic phonon coupling revealed by resonant excitation of single perovskite nanocrystals

Single perovskite nanocrystals have attracted great research attention very recently due to their potential quantum-information applications, which critically depend on the development of powerful optical techniques to resolve delicate exciton photophysics. Here we have realized resonant and near-resonant excitations of single perovskite CsPbI3 nanocrystals, with the scattered laser light contributing to only ~10% of the total collected signals. This allows us to estimate an ultranarrow photoluminescence excitation linewidth of ~11.32 µeV for the emission state of a single CsPbI3 nanocrystal, corresponding to an exciton dephasing time of ~116.29 ps. Meanwhile, size-quantized acoustic phonons can be resolved from a single CsPbI3 nanocrystal, whose coupling with the exciton is proposed to arise from the piezoelectric potential. The ability to collect resonance fluorescence from single CsPbI3 nanocrystals, with the subsequent revelation of exciton-acoustic phonon coupling, has marked a critical step towards their steady advancement into superior quantum-light sources.


Supplementary Methods
Chemical synthesis. To prepare the Cs-oleate precursor, Cs2CO3 (0.54 g), oleic acid (OA, 1.7 mL) and octadecene (ODE, 20 mL) were loaded into a 100 mL three-neck flask and dried under vacuum for 1 h at 120 ℃. The solution was then heated to 150 ℃ under N2 until it became clear. For the synthesis of the CsPbI3 NCs, PbI2 (0.174 g), ODE (10 mL), OA (0.5 mL) and oleylamine (OLA, 0.5 mL) were loaded into a 50 mL three-neck flask and dried under vacuum for 1 h at 120 ℃. The temperature of this mixture was raised to 160 ℃ subsequently, into which the preheated Cs-oleate solution (0.8 mL) was swiftly injected. After 5 s, the reaction mixture was transferred to the ice-water bath and 1 mL of tri-octylphosphine was injected at the temperature of 80 ℃. To purify the CsPbI3 NCs, the resulting solution was centrifuged for 5 min at 4000 rpm and the suspension was further centrifuged for 10 min at 10000 rpm. Finally, the precipitate was dispersed in hexane and stored in the glove box.
Optical measurements. One drop of the diluted NC solution was spin-coated onto a fused silica substrate, where only several bright spots could be detected from the confocal scanning PL image measured for a sample area of 10 µm × 10 µm. Such a bright spot corresponds to the optical emission from a single CsPbI3 NC, as confirmed previously from the photon antibunching measurement in ref. 5 of the main text. The sample substrate was attached to the cold finger of a helium-free cryostat, while a He-Ne laser and a tunable diode laser both operated at the continuous-wave mode were employed for the above-bandgap and resonant/near-resonant excitations of a single CsPbI3 NC, respectively. The laser beam was focused onto the sample substrate by a dry objective with a numerical aperture of 0.82 and the optical signal collected by the same objective was sent to a spectrometer (0.75 m, 1200 g/mm grating) and a chargecoupled device (CCD) camera for the spectral measurement with a resolution of ~100 µeV.
For the purpose of observing resonance/near-resonance fluorescence from a single CsPbI3 NC 3 excited by the tunable diode laser, two Glan-Thompson polarizers with orthogonal transmission axes were inserted into the laser excitation and PL collection paths, respectively.
A quarter-wave plate was mounted to a stage with a rotating precision of 0.01 degree to correct the birefringence effect caused by the relevant optical components. To further suppress the scattered laser light, the entrance slit of the spectrometer was closed to ~120 µm for the spatial filtering and an extinction ratio exceeding 10 5 could be obtained at a typical laser wavelength of 725 nm. As for the above-bandgap excitation, a band-pass optical filter was used to separate the PL signal of a single CsPbI3 NC and the scattered light from the He-Ne laser. Unless otherwise specified in the text, all the optical measurements were performed at the cryogenic temperature of 3 K, and the laser excitation power was normally set at ~100-500 nW so that the PL intensity of a single CsPbI3 NC was not saturated to minimize the possibility of generating multiple excitons.

Theoretical Calculations
Modelling details. The acoustic waves in a solid satisfy the following differential equation [ref. S1], where , is the lattice displacement, is the stiffness tensor, and is the mass density.
For a plane wave, the lattice displacement can be written as where is the polarization vector and = −1 is the slowness vector, with and being the sound velocity and the wavefront-normal vector, respectively. The sound velocity can be obtained by solving the eigen-value Christoffel equation, By using the Voigt notation, the fourth-order stiffness tensor can be transformed into a second-order 6 × 6 tensor . For a material with the tetragonal symmetry, has the form of  Piezoelectric coupling. Since the phonons could cause the displacement and hence the strain field, we would expect a piezoelectric coupling between the electrons and acoustic phonons in a single CsPbI3 NC. This strain, via the piezoelectric tensor ; , is able to create an electric field that interacts with the electron or hole. The electric displacement component (k = x, y or z) in the presence of the piezoelectric effect can be described by  the result of a strong electromechanical coupling due to its ionic feature, and a low speed of sound due to its elastic softness for the transverse modes.
Emission and absorption spectra. We model the emission and absorption spectra of single CsPbI3 NCs by explicitly including the effect of acoustic-phonon scattering. For clarity, we consider a single exciton state that couples to a phonon bath but note that the formalism can be 7 readily extended to the case with multiple discrete exciton states. The exciton-phonon Hamiltonian can be written as where | ⟩ and 0 are the exciton state and its energy. The second term describes the phonon bath with + ( ) creating ( where represents phonon mode ( ) and the summation over phonon number is from −∞ to +∞. The -function of energy conservation can be replaced by a Lorentzian function where is the exciton broadening. The optical absorption spectrum can be similarly expressed as Since both and ′ decrease with | | rapidly, the emission and absorption spectra can be adequately simulated by considering only | | < 3.
In Supplementary Fig. 7, we display the PL intensity as a function of the detuned energy ℏ − 0 at = 3 K, which is calculated for the PL spectrum shown in Fig. 4a of the main text from a single CsPbI3 NC under resonant excitation of its higher-energy peak. We consider a strong piezoelectric coupling with TA phonons (ℏ = 155 µeV) and a relatively weak piezoelectric/deformational coupling with LA phonons (ℏ = 505 µeV), with their dimensionless coupling strengths being assumed to be | | = 0.06 and 0.02, respectively. The exciton broadening is set to be 17.6 µeV that is consistent with the experimental result. We see that the one-phonon peak of 155 µeV emerges both above and below the ZPL. The peak below the ZPL, which corresponds to the phonon emission, has a higher intensity because the emission (absorption) of phonon is proportional to + 1 ( ) and the difference can be significant at low temperatures when ≪ 1. This also explains that a pronounced onephonon peak of 505 µeV appears only below the ZPL, which, however, is not resolvable from 9 the background due to the limited signal-to-noise ratio in our experiment. Two and higher phonon peaks for both 155 µeV and 505 µeV are too weak to be clearly distinguished from the natural broadening. It is remarkable that our model with only two phonon modes can account for the measured PL spectrum very satisfactorily.
The calculated absorption spectra for a single CsPbI3 NC are plotted in Supplementary Fig. 6, which are related to the PL excitation spectra shown in Fig. 3b of the main text. The measured PL excitation spectra at 3 and 10 K can be well accounted for by considering the two TA (190 µeV) and LA (626 µeV) phonon modes with the dimensionless coupling strengths of | | = 0.06 and 0.02, respectively. It is interesting to note that in the absorption spectrum, it is the phonon peaks above the ZPL that are more pronounced. At = 3 K, the exciton broadening is set to be 17.6 µeV and it becomes 64 µeV at = 10 K. The larger broadening at = 10 K makes the discrete phonon peaks less pronounced. charge-coupled device. The laser beam first passes through a linear polarizer and is then reflected by a PBS, which are respectively chosen to transmit and reflect light with the vertical linear polarization. After being focused by the objective and scattered by the sample substrate, the laser light collected by the same objective is blocked first by the PBS and then by a horizontal linear polarizer placed before the spectrometer and the CCD camera. To minimize the residual laser light that still arrives at the CCD camera, a QWP is inserted between the PBS and the objective to correct the birefringence effect caused by relevant components in the optical path. Under resonant excitation of the tunable diode laser, a single CsPbI3 NC is always chosen to emit doublet peaks with comparable PL intensities, implying that the two transition dipole moments are aligned at ~45º relative to the laser polarization direction. In this case, part of the doublet-peak photons emitted by a single CsPbI3 NC can still arrive at the CCD camera for the resonance/near-resonance fluorescence measurement, after passing through the PBS and the horizontal linear polarizer sequentially.