Quantifying the Size‐Dependent Exciton‐Phonon Coupling Strength in Single Lead‐Halide Perovskite Quantum Dots

Optimizing the performance of semiconductors in both classical and quantum applications, not only requires a solid understanding of elementary excitations such as electrons, holes, or bound electron–hole pairs (excitons), but also of their interaction with the host material's vibrational states (phonons). Exciton‐phonon coupling is particularly relevant in quantum dots (QDs) of APbX3 lead‐halide perovskite (where “A” can be Cs, formamidinium (FA), or methylammonium (MA), and X can be Cl, Br, or I), a new class of semiconductors with a soft crystal structure. Here, they quantify the strength of coupling to interband transitions for both FAPbBr3 and CsPbBr3 QDs, via the magnitude of phonon replicas in their photoluminescence (PL) spectra at cryogenic temperatures. CsPbBr3 QDs exhibit weaker exciton‐phonon coupling than similarly sized FAPbBr3 QDs. While the phonon energies are size‐independent, the exciton‐phonon coupling strength decreases with increasing QD size due to the decreased coupling of the transition to low‐energy surface‐enhanced phonon modes, consistent withab initio molecular‐dynamics (AIMD) simulations. Furthermore, within the harmonic approximation, the size‐dependent PL linewidth at room temperature can coarsely be estimated from the low‐temperature phonon replica spectrum, highlighting the crucial role of anharmonic effects. These findings contribute to realizing perovskite QD‐based devices with narrow and coherent emission for quantum technologies.


Introduction
Lead-halide perovskite (APbX 3 ) quantum dots (QDs) have attracted much research interest due to compelling optoelectronic properties, [1,2] facile synthesis [2][3] with precise control of QD size and shape, [3][4] wide spectral tunability, [5] high photoluminescence (PL) quantum yield, high oscillator strength, [6] and single-photon emission. [7]These QDs also became a promising building block for applications such as LEDs with external quantum efficiencies above 20%, [5,8] lasers with wide spectral tunability, [9] photodetectors with record-high responsivity, [10] and sources of quantum light. [7,11]The APbX 3 perovskite crystal structure is characterized by a threedimensional network of corner-sharing PbX 6 octahedra (X = Br, Cl, I), with the A-site cation filling the voids.As shown in Figure 1a, the A-site cation is either Cs + , an inorganic atom, or an organic molecule such as CH(NH 2 ) 2 + (formamidinium, FA + ).At cryogenic temperature, both FAPbBr 3 and CsPbBr 3 QDs exhibit orthorhombic crystal structure.To allow synthetic efforts to further optimize the already outstanding luminescent properties of APbBr 3 QDs, the exciton recombination process should be understood in detail.
A key aspect of the latter is exciton-phonon coupling as it impacts fundamental processes such as homogeneous emission broadening and excitonic dephasing.For example, color-pure LEDs require ultra-narrow emission at room temperature (RT) and, hence, materials with reduced exciton-phonon coupling.7b,11a] Exciton-phonon coupling is particularly rich in lead-halide perovskite QDs due to the many vibrational degrees of freedom and the soft character of the multi-component host lattice.Exciton-phonon coupling in lead-halide perovskite QDs was previously investigated by means of Raman spectroscopy, [12] 2D electronic spectroscopy, [13] ab initio molecular-dynamics (AIMD) calculations, [1,14] and low-temperature single-QD spectroscopy. [15]The latter method is particularly powerful, as it i) directly probes the emission process, ii) distinguishes coupling to a broad range of phonon modes, and iii) is free of inhomogeneities inherent to many ensemble-based studies, as it probes intrinsic emission properties of single QDs.With respect to phonon-mediated emission, each phonon mode may give rise to phonon sidebands, occurring at energies shifted by ± kℏ with respect to the zero-phonon line (ZPL), where k is the number of phonons emitted/absorbed in the transition and ℏ is the phonon energy.The side-band intensities I k follow a Poisson distribution as where I 0 the intensity of the ZPL, and S is the Huang-Rhys factor characterizing the exciton-phonon coupling strength to the transition.Experimentally, the latter is typically inferred from the intensity ratio of the first phonon replica to the ZPL, as S = I 1 /I 0 .
In general, this methodology can be broadly applicable to systems of all dimensionalities, including QDs, nanowires/nanorods, nanoplatelets, layered materials, etc.Previously, such a methodology has successfully been applied also in studies of CdSe QDs [16] and InP QDs. [17]In practice, a main limitation concerns the spectral overlap of the phonon replica spectrum with exciton complexes, such as trions and biexcitons, and limited signal intensity from phonon replicas.To prevent contributions from exciton complexes, QDs should be excited at very low excitation density.Long integration times are required to increase the signal-to-noise ratio.Importantly, the latter also requires that the studied QDs are stable for the duration of the measurement.
In this work, we investigate the size-dependent excitonphonon coupling by monitoring PL spectra of FAPbBr 3 and CsPbBr 3 QDs at the single-particle level at 4 K.Consistent with earlier investigations using AIMD simulations, [1] the increase of the coupling strength with decreasing QD size is especially pronounced for low-energy phonon modes.Similar qualitative trends may be found in nanomaterials of other chemical compositions and shapes, see e.g., PbS QDs. [18]Here, we now further elucidate the microscopic nature of these modes via additional DFT calculations.To relate the exciton-phonon coupling strength extracted at cryogenic temperature to the RT homogeneous PL line broadening, we attempt to estimate the latter based on the extracted effective coupling strength within the harmonic approximation.Such a simple estimate reproduces the well-documented size-dependent PL line broadening, [1,7a,19] affirming our assignment of coupling dominated by a deformation-potential mechanism.However, the magnitude of the broadening is underestimated within the harmonic approximation, highlighting the potential role of anharmonic effects in soft perovskite compounds at elevated temperatures. [20]

Exciton-Phonon Coupling in Single FAPbBr 3 QDs
FAPbBr 3 QDs with an edge length of 9 ± 1 nm (Figure S1a, Supporting Information) and a RT ensemble PL central wavelength of 525 nm (2.3638 eV) (see Figure 1b) were synthesized according to a previously published protocol. [3]15a,d,21] A typical spectrum of a single FAPbBr 3 QD is shown in Figure 2a, with the emission energy referenced to the energy of the ZPL (2.2595 eV).Three distinct peaks are observed and fit with a multi-Lorentzian function (for details see Equations S1-S3, Supporting Information).The ZPL at 0 meV corresponds to the exciton recombination whereas the phonon replicas at −4.7 and −19.5 meV are assigned to redshifted PL involving the emission of optical phonons (OP) with energies equal to the modulus of the shift.We here denote these phonons for simplicity as OP 1 and OP 2 , respectively, while noting that due to line broadening mechanisms and a finite spectral resolution (vide infra), we cannot exclude the involvement of more than only two phonons, if closely distributed around the "effective" energies 4.7 and 19.5 meV, respectively.The emergence of such OP 1 and OP 2 phonon replicas in the low-T PL spectra reflects significant coupling of the exciton to these two effective phonon modes and suggests that these modes also affect excitonic dephasing and RT homogeneous linewidths.Furthermore, both the ZPL and phonon replicas appear to be broadened homogeneously via coupling to phonons, as inferred from the unresolved exciton fine-structure splitting of the ZPL [22] (see Figure S2a, Supporting Information), despite sufficient spectral resolution of the experimental setup (0.2 meV).The spectrum in Figure 2a features a continuous emissive band within the range from ≈8 to ≈18 meV, which we exclude from our fitting.Omitting this part has a negligible effect on the extracted exciton-phonon coupling (see Section SII, Figure S3a, and Table S1, Supporting Information).This band may stem from unresolved higher-order peaks of OP 1 as discussed in ref.
[15a] or from coupling to additional phonon modes within the broad phonon density of states of FAPbBr 3 .
Exciton and phonon energies of all studied single FAPbBr 3 QDs are summarized in Figure 2b.The emission energies cover the range of 2.24-2.31eV and correspond to 6-13 nm QDs according to the sizing curve from ref.
[15a].This size range agrees well with the size distribution obtained from the TEM images (see Figure Supporting Information).The phonon energies in Figure 2b are independent of the QD size, as previously observed for perovskite QDs [15a,d,23] and non-perovskite QDs, [24] and narrowly distributed around 4.9 meV and 19.6 meV, respectively.As confirmed via calculation, the OP 1 mode involves Pb─X─Pb bond-angle distortion while the OP 2 mode is dominated by the stretching of the Pb─X─Pb bond. [14]he Huang-Rhys factor S describes the exciton-phonon coupling strength which is experimentally obtained as the intensity ratio of the first phonon replica and the ZPL.Consistent with previous studies, [15a] Figure 2c shows that the coupling strength increases for smaller QDs.The size dependence is especially pronounced for the low-energy mode OP 1 , for which S increases from 0.1 to 0.8 for exciton emission between 2.24 and 2.31 eV; in contrast, for the OP 2 mode, S increases only from 0.05 to 0.09 in the same energy range.

Exciton-Phonon Coupling in Single CsPbBr 3 QDs
After quantifying exciton-phonon coupling in FAPbBr 3 QDs, we employ the same methodology to CsPbBr 3 QDs.We studied three different QD samples with sizes of 9, 13, and 20 nm (Figure S1, Supporting Information), with RT ensemble PL wavelengths of 514 nm (2.4144 eV), 516 nm (2.4050 eV), and 520 nm (2.3865 eV), respectively (Figure 1b). Figure 3a shows the 4 K PL spectrum of a representative single QD from the ensemble with a mean QD size of 13 nm; the 4 K emission energy is referenced to the ZPL (2.3576 eV).PL spectra for CsPbBr 3 QDs were integrated over 100 s to resolve phonon replicas, enabled by the good spectral stability of the studied single QDs (see Figure S2, Supporting Information) with the observation of three phonon replicas denoted as OP 1 , OP 2 , and OP 3 , with energies of 3.5, 6.2, and 19.0 meV, respectively.We also resolve the characteristic fine structure of the ZPL and phonon replicas -one, two, or three emission lines from a bright triplet exciton [7e] -indicating a weaker homogeneous broadening of the ZPL and phonon replicas than for FAPbBr 3 QDs (see Figure 2a).For the spectrum of the particular QD shown in Figure 3a, doublet emission lines for ZPL and corresponding phonon replicas are observed, each with a fine-structure splitting of 0.8 meV.
For single QDs across a large size range, we consistently observe phonon modes at mean energies of 3.5, 6.3, and 19.0 meV, respectively, as shown in Figure 3b.12a,13a,15d,f] Its absence in large QDs (20 nm; Figure S4a, Supporting Information) suggests that the excitonic coupling to this mode reduces strongly with QD size.Omitting this mode from the analysis did not affect the deduced exciton-phonon coupling strength at any particle size (see Section SII, Figure S3b, and Table S2, Supporting Information).Similarly in FAPbBr 3 QDs, phonon energies are size-independent (see Figure 3b).Comparing Figure 2b and Figure 3b, the phonon replicas appear less broad in CsPbBr 3 than in FAPbBr 3 , which may arise from stronger spatial and temporal fluctuations in FAPbBr 3 QDs.Such fluctuations, persisting even at cryogenic temperatures, have previously been reported in both FAPbBr 3 [25] and FAPbI 3 QDs. [26]Thus, our observation of broad phonon-replica spectra in FAPbBr 3 QDs (but not CsPbBr 3 QDs) is consistent with these earlier reports and may also explain that only one low-energy phonon replica could be resolved in these compounds.
Next, the coupling strength (S = I 1 /I 0 ) is plotted in Figure 3c.For successively smaller QDs with ZPL emission energy increasing from 2.32 to 2.39 eV, S exhibits a strong increase from 0.025 to 0.2 for the low-energy mode OP 1 , a moderate increase from 0.015 to 0.1 for the medium-energy mode OP 2, and a weaker increase from 0.007 to 0.03 for the high-energy mode OP 3 , respectively.Compared to the FAPbBr 3 QDs, a weaker homogeneous broadening of the ZPL, lower intensity ratio of phonon replicas, and the absence of features from higher-order phonons indicate an overall weaker exciton-phonon coupling in CsPbBr 3 QDs.The fraction of emission into phonon replicas compared to ZPL emission is below 10% for the large QDs (20 nm) suggesting that over 90% of photons are emitted from the ZPL transition due to the highly suppressed coupling to phonons.7b,11a] We have previously shown via RT single-QD PL spectroscopy and AIMD simulations that the exciton-phonon coupling strength is enhanced in smaller QDs due to enhanced coupling to large-amplitude surface phonon modes. [1]This low-temperature single-QD study now corroborates this earlier finding, showing that with decreasing QD size, lower-energy phonons (OP 1 and OP 2 in CsPbBr 3 QDs and OP 1 in FAPbBr 3 QDs, respectively) couple progressively stronger to the electronic transition.

Size-Dependent Coupling to Low-Energy Optical Phonons in Lead-Halide Perovskites via the Deformation Potential
In Figure 4a, we plot the phonon density of states of CsPbBr 3 computed at the DFT level of theory, along with the emission spectra of CsPbBr 3 QDs of three different sizes normalized to the integrated intensity of the ZPL in Figure 4b.The higher-energy optical phonon mode which couples to the exciton at ≈19 meV in both CsPbBr 3 and FAPbBr 3 QDs corresponds to a mode that drives Jahn-Teller-like tetragonal compressions/elongations of the Pb-halide octahedra, where the Br ions are displaced in directions parallel to the Pb─Br bonds. [27]Coupling to this mode has received considerable attention in the lead-halide perovskite community. [28]12a,28b,c,29] However, several inconsistencies arise with this assumption.First, recent experimental observations of transient lattice ordering in APbBr 3 QDs upon photoexcitation [14,30] are in stark contrast to the symmetry-lowering polar distortions driven by the Fröhlich interaction.Additionally, such a polar interaction would require net charge-density variations over the length scale of the exciton (on the order of the QD size); however, the reported fast radiative decay at low temperatures [7e] is inconsistent with an assumption of significant electron-hole separation and localization.12a] Lastly, our low-temperature PL spectra indicate only a weak coupling to the ≈19 meV mode (OP 3 ) and a much stronger coupling to the lower energy optical modes (OP 1 and OP 2 ).
Here, we suggest that exciton-phonon coupling occurs largely via the deformation-potential mechanism, where lattice reorganization minimizes the exciton energy through band-gap renormalization, as opposed to a polar Fröhlich distortion which minimizes long-ranged Coulomb interactions.As shown in Figure 4c, the lower-energy optical phonon modes in the range of ≈3-7 meV correspond to oscillations of the Pb─Br─Pb bend angles between the lead-halide octahedra.With the LHP bandgap sensitive to these angles, [14] straightening of the Pb-Br-Pb bend angles during photoinduced lattice reorganization yields a transient band gap reduction.Frozen-phonon calculations on CsPbBr 3 (see Section SIII, Supporting Information) reveal two modes at 3.7 and 6.2 meV that couple strongly as a result of this mechanism (vertical lines in Figure 4a [14] and shown in Figure 4c).The energies of these modes match very closely with the peak energies of the phonon sidebands (OP 1 , OP 2 ) in the low-temperature PL spectra of CsPbBr 3 QDs.Finally, our experimentally observed increased exciton-phonon coupling strength for smaller QDs is also consistent with a deformation-potential coupling mechanism, as the coupling strength of the latter, to first order, scales with the volume of the exciton V (on the order of the QD volume) as S  ∝V −1 . [14,31]

Relation to the Room-Temperature PL Linewidth
Exciton-phonon coupling determines the homogenous PL linewidth.Within the harmonic approximation and considering a relatively strong exciton-phonon coupling, the homogeneous PL linewidth at temperature T is given by where Λ is the reorganization energy and k B the Boltzmann constant.Λ is the product of the effective Huang-Rhys factor S eff and the effective phonon energy ℏ eff which can both be extracted from PL spectra (Equations S4 and S5, Supporting Information).Figure 5a reports Λ for single QDs across a broad size range.With decreasing QD size, Λ systematically increases suggesting a larger linewidth for smaller QDs.
To explore the relevance of the exciton-phonon coupling strength extracted from phonon-replica spectra at 4 K for homogeneous broadening at higher temperatures, we performed temperature-dependent single QD measurements on the same batches.As an example, Figure 5b displays the temperaturedependent line widths (full width at half maximum, FWHM) for a single 13 nm QD from 4 K up to 250 K.As shown in the inset of Figure 5b, the PL peak blueshifts and broadens with increasing temperature, due to lattice expansion and exciton-phonon coupling, respectively.Quantitatively, the linewidth increases from 2 meV (limited by the spectral resolution) at 4 K to 65 meV at 250 K.
Figure 5c displays the measured RT linewidth of several single QDs as a function of their exciton energy.7a,19] Figure 5d shows a firstorder estimate of the RT linewidths estimated from the values of Λ extracted from the 4 K phonon-replica spectra and utilizing the harmonic approximation underlying Equation (2) and Equation S6 (Supporting Information).This estimated RT linewidth is shown as a function of the expected RT energy (the RT exciton energy is roughly 60 meV blueshifted compared to the exciton energy at 4 K, see Figure S5, Supporting Information).The predicted RT linewidth increases from 10 to 40 meV for QD sizes decreasing from 20 to 9 nm.Hence, qualitatively, the size dependence of the RT linewidth predicted from the cryogenic phonon replica spectra agrees well with the linewidth measured at RT. Quantitatively, however, the predicted linewidths are systematically underestimated by ≈40 meV.Some underestimation is unsurprising: i) our simple estimate (given by Equation 2) is only exact for strong excitonphonon coupling and for high temperature and does not yet include temperature-dependent damping mechanisms [32] which would broaden each line, i.e., ZPL and replicas, [33] and yield a steeper temperature dependence; indeed, thermally induced damping is expected from the reported short and temperaturedependent phonon lifetimes in lead-halide perovskites; [34] ii) additional anharmonic contributions may become unlocked at higher temperature, [16][17]32,35] e.g., temperature-dependent crystal structure variations, surface mobilities due to dynamic ligands, further increasing the coupling strength and broadening; iii) the experimental ZPL contribution at 4 K may be overestimated due to resolution-limited spectral diffusion and homogeneous broadening, yielding underestimated coupling strengths and RT linewidths; However, quantifying the precise magnitude of the various anharmonic contributions to the temperature-dependent PL broadening is beyond the aim and scope of this work. Neverteless, single-QD phonon replica spectra acquired at 4 K return important insights into the mechanism of exciton-phonon coupling as well as the nature and frequency of involved phonons while providing a first-order estimate for the size-dependent linewidth at RT within the harmonic approximation.

Summary
In summary, by probing the phonon replicas in single-QD PL spectra at 4 K, we quantified the exciton-phonon coupling strength arising from at least two and three dominant optical phonon modes in FAPbBr 3 and CsPbBr 3 QDs, respectively.For both material systems, lower-energy phonon modes are sensitive to the QD surface and hence their coupling strengths to excitons are enhanced in small QDs.Generally, exciton-phonon coupling is stronger in FAPbBr 3 QDs than in CsPbBr 3 QDs.Utilizing an experimentally obtained effective exciton-phonon coupling strength at cryogenic temperature, we present a first-order estimate of the homogeneous room-temperature PL linewidth within the harmonic approximation and coarsely recover the size-dependent trend of the PL linewidth observed in experiments.A systematic underestimation of the latter may point to the existence of anharmonic contributions at elevated temperatures.Our studies on the size-dependent exciton-phonon coupling will facilitate the progressive optimization of low-cost leadhalide-perovskite-based devices requiring narrow emission and enhanced coherence for quantum technologies.

Figure 1 .
Figure 1.Structural and optical properties of APbBr 3 QDs.a) Crystal structure (orthorhombic at cryogenic temperature) of CsPbBr 3 and FAPbBr 3 QDs.b) The room-temperature ensemble PL spectra of the FAPbBr 3 and CsPbBr 3 QDs studied in this work, with mean sizes as indicated in the legend.

Figure 2 .
Figure 2. Exciton-phonon coupling in single FAPbBr 3 QDs.a) Representative PL spectrum of a single FAPbBr 3 QD (from a QD ensemble with a mean QD size of 9 nm) at 4 K.The ZPL and two phonon replicas are fitted by Lorentzian peaks.b) Phonon and exciton energy distribution extracted from PL spectra of several single QDs.Upper panel: histogram of phonon energies.Lower panel: size independence of optical phonon energies.c) The positive correlation of the exciton energy and the intensity ratio of phonon replicas to ZPL reveals increasing exciton-phonon coupling strength for decreasing QD size.Dashed lines are guides to the eye.

Figure 3 .
Figure 3. Exciton-phonon coupling in single CsPbBr 3 QDs.a) Representative PL spectrum of a single CsPbBr 3 QD (from a QD ensemble with a mean QD size of 13 nm) at 4 K.The ZPL and three optical phonon replicas are fitted by Lorentzian peaks.b) Phonon and exciton energy distribution extracted from PL spectra of single QDs from all three samples.Upper panel: histogram of phonon energies.Lower panel: size independence of optical phonon energies.c) The positive correlation of the exciton energy and the intensity ratio of phonon replicas to ZPL reveals increasing exciton-phonon coupling strength for decreasing QD size.Results are from all three samples.Dashed lines are guides to the eye.

Figure 4 .
Figure 4. Coupling of excitons to lower-energy optical phonons in lead-halide perovskites a) Computed phonon density of states of bulk CsPbBr 3 and b) experimentally measured spectrum of single CsPbBr 3 QDs of different sizes at 4 K; the intensity of each spectrum is normalized to the integrated intensity of the ZPL (not shown here).c) Visualization of the OP 1 and OP 2 optical phonon modes.The length of the arrows corresponds to the magnitude of the atomic displacement in the mode (displacements smaller than 20% of the maximum displacement are not shown).

Figure 5 .
Figure 5. Homogeneous PL broadening analysis for CsPbBr 3 QDs.a) Distribution of the reorganization energy (Λ) obtained from the phonon replica spectra based on Equation S4 (Supporting Information), for single QDs of batches with a mean size as indicated in the legend.b) Temperature-dependent linewidth (FWHM) of a single 13 nm CsPbBr 3 QD at temperatures from 4 to 250 K. Inset: associated PL spectra at different temperatures.c) Experimentally measured linewidth (FWHM) of single QDs at room temperature (RT), increasing with increasing exciton energy, i.e., decreasing size.d) Predicted FWHM at RT based on Equation (2).