Microscopic Picture of Electron–Phonon Interaction in Two-Dimensional Halide Perovskites

Perovskites have attracted much attention due to their remarkable optical properties. While it is well established that excitons dominate their optical response, the impact of higher excitonic states and formation of phonon sidebands in optical spectra still need to be better understood. Here, we perform a theoretical study of excitonic properties of monolayered hybrid organic perovskites—supported by temperature-dependent photoluminescence measurements. Solving the Wannier equation, we obtain microscopic access to the Rydberg-like series of excitonic states including their wave functions and binding energies. Exploiting the generalized Elliot formula, we calculate the photoluminescence spectra demonstrating a pronounced contribution of a phonon sideband for temperatures up to 50 K, in agreement with experimental measurements. Finally, we predict temperature-dependent line widths of the three energetically lowest excitonic transitions and identify the underlying phonon-driven scattering processes.

I n the past years, much effort has been devoted to the study of perovskite materials. 1−5 Their broad absorption spectrum 6 combined with impressive light-emitting properties 7 makes them a promising material for optoelectronic applications. They have been, in particular, applied as active material in solar devices exhibiting a power conversion efficiency of up to 25.2%. 8 Beyond photovoltaics there has been a rising interest in perovskite-based diodes and lasers. 9−12 Recently, layered perovskites have emerged as a subclass of the bulk material exhibiting enhanced excitonic properties and tunable characteristics. 13 Their optical and electronic properties can be tailored through an external stress 14,15 or an internal chemical pressure 16 or by varying their layer thickness. 17 In addition, a wide variety of compositions can be made thanks to the large amount of organic cations that can be integrated into the structure. These materials are considered as a dimensional reduction of 3D perovskites with the general formula ′ − + (A) A B X m n n n 1 3 1 , where A′ is the cation that intercalates between the inorganic A n−1 B n X 3n+1 layers and n can be understood as the thickness of the inorganic layers. Here, the screening of the Coulomb interaction is weaker compared to 3D perovskites, resulting in considerably larger exciton binding energies. 18,19 As a direct consequence, excitons play a crucial role in the optical response and nonequilibrium dynamics of these materials. Unbound carriers can still play a significant role at higher densities and excitations above the single-particle bandgap. Recent experimental studies on optical properties of 2D perovskites have focused on exciton binding energies, 19 absorption line widths, 12,20−24 phonon sidebands, 25 and high-density effects. 26 However, a microscopic understanding of some of the reported findings, for example, the formation of phonon sidebands and the microscopic origin of excitonic line widths, has not been well understood yet.
Here, we present a joint theoretical study of optical properties of monolayered hybrid organic perovskites supported by experimental photoluminescence (PL) measurements. The aim of the work is to provide a microscopic understanding of many-particle processes determining the temperature-dependent PL in these materials. We focus on the monolayered (PEA) 2 PbI 4 , which is a particular case of the nlayered (PEA) 2 MA n−1 Pb n I 3n+1 perovskite 27,28 (cf. Figure 1a). Here, the inorganic layer consists of corner-sharing PbI 6 octahedra sandwiched between two organic layers of PEA (phenylethylammonium). The lower dielectric constant of the organic layer results in a dielectrically confined quantum well.
Using a fully quantum mechanical approach, we obtain microscopic access to excitonic wave functions and binding energies in 2D perovskites. Exploiting the generalized Elliot formula, we model temperature-dependent PL spectra including the emergence of phonon-assisted sidebands. Finally, we determine the spectral line width of the energetically lowest excitonic transitions taking into account scattering channels driven by the emission and absorption of acoustic and optical phonons (cf. Figure 1b). The theoretical predictions are compared to temperature-dependent measurements of photoluminescence spectra in the investigated 2D perovskites.
As a first step, we determine the excitonic properties of monolayered (PEA) 2 PbI 4 perovskites. To obtain microscopic access to the Rydberg-like series of excitonic states ν, their binding energies ν E b , and wave functions ϕ ν k , we solve the Wannier equation 29 where we approximate the Coulomb interaction by the to the treatment of atomically thin transition metal dichalcogenides (TMDs). 34 Here, we used the exciton reduced mass 21 μ = 0.108m 0 and the average dielectric constant 35 ε s = 3.32 given by the organic layers. The dielectric constant of the inorganic layer is ε well = 6.1, and its thickness is L well = 0.636 nm. 35 These quantities determine the screening length = ε ε r L 0 w e l l 2 well s , which is crucial for the excitonic binding energy Evaluating the Wannier equation, we find = E 228 b 1s meV for the energetically lowest 1s exciton state, while the binding energy decreases to 43 and 17 meV in the case of 2s and 3s states, respectively (cf. Figure 2a and Table 1). The obtained exciton binding energies are in good agreement with experimentally measured values (cf. the inset of Figure 2a). These experimental values have been determined by identifying the spectral position of the exciton resonances (1s and 2s) and the single-particle band edge in photoluminescence excitation, 25 optical absorption, 24,35,36 and electroabsorption 37 measurements or by extracting the thermal dissociation ratio of excitons from temperature-dependent photoluminescence measurements. 38 Note that interestingly the 2p state has a larger binding energy than the corresponding 2s state similarly to the case of TMD monolayers. 39 The corresponding eigenfunctions of the three lowest excitonic states are illustrated in Figure 2b and show the typical momentum dependence well known from the hydrogen problem.
The obtained excitonic binding energies and wave functions will be used in the following to calculate the photoluminescence spectrum and the line widths of excitonic transitions in the investigated 2D perovskite material.
The photoluminescence is calculated by applying the density matrix formalism for an interacting system of electrons, phonons, and photons. 40 . 31,43 Here, radiative and nonradiative decay channels play the crucial role. The radiative recombination of exciton states within the light cone is given by 44,45     , and the optical matrix element for interband transitions M σ = 8 fs eV nm −1 projected into the polarization σ of the emitted light. In the considered low-excitation regime, phonon-driven scattering channels determine the nonradiative dephasing 40,44 . The excitonic dispersion ν E k is described in an effective mass approximation with a total exciton mass M = 0.44m 0 . 21 The equation includes absorption and emission (±) scattering channels involving acoustic and optical phonons with the mode j. The emission term (+) includes spontaneous and stimulated processes. The latter is proportional to the phonon occupation n j q that is given by the Bose−Einstein distribution within the considered bath approximation.
The appearing exciton−phonon matrix element is given by the coupling elements for electrons and holes g e/h sandwiched by the exciton wave functions ϕ μ k of the involved initial and final exciton states with α λ = m λ /M. 44,45 The electron−phonon for acoustic phonon modes. 46,47 The deformation potential interaction is equivalent to applying pressure or strain to the system, 47 resulting in energy shifts in opposite direction for conduction and valence bands. These energy shifts are reflected in the coupling strengths g λ (note the ± signs in the deformation potentials defined above), and G is thus a result of adding the conduction (g c ) and valence (g v ) band terms. While the Froḧlich interaction dominates the electron−phonon interaction, its polar nature results in g c and g v having the same sign (as the energies shift in the same direction) and hence a weak exciton−phonon coupling strength G. First-principles studies are needed to obtain access to the full phonon dispersion and electron−phonon coupling elements as well as to thoroughly describe the importance of deformation potential and Froḧlich interactions. While this is beyond the scope of this work, we extract the values for deformation potentials from the experimentally measured temperature-dependent line width of the 1s exciton assuming one dominant optical and acoustic phonon branch (cf. Table  2). Note that these are effective modes that can contain several spectrally close phonon branches. We find that the linear increase of the PL line width at low temperatures can be traced back to the scattering with acoustic phonons characterized by the velocity c AC = 2100 m/s. At high temperatures, there is a deviation from a linear increase and the major contribution stems from scattering with optical phonons exhibiting an energy of 35 meV, similar to recent experimental findings. 21,36,48 Note that there is also a defect-induced line width Γ 0 = 6.7 meV at zero temperature. Considering a many-particle Hamilton operator including exciton−photon and exciton−phonon interaction and evaluating the Heisenberg equation of motion for the photon number n k , we find a coupled set of differential equations including phonon-and photon-assisted polarizations. Then, the cluster expansion scheme 49 is used to factorize the many-particle expectation values disregarding contributions connected to multiphonon processes. 40 The final analytic expression for the σ-polarized photon flux emitted in perpendicular direction with respect to the monolayer is given by 40 where η = ± + ± n 1/2 1/2 j j q q denotes the relevant phonon occupation factor for absorption/emission and ν N q represents the exciton occupation corresponding to the Boltzmann distribution for the considered stationary PL. The first term describes the direct photoluminescence that is often termed zero-phonon contribution. The second term shows the phonon-assisted indirect PL and results in the emergence of phonon sidebands. Evaluating eq 3, we obtain a microscopic access to the temperature-dependent PL and find that the line width of the exciton transition clearly increases with temperature, while its intensity becomes smaller (cf. Figure 3a). The zero-phonon line width ranges between ∼5−10 meV at low temperatures and ∼50 meV at room temperature.
In Figure 3b−d, we decompose the direct PL stemming from radiative exciton recombination within the light cone (Q ≈ 0) from the indirect contribution to the PL arising from phonon-assisted exciton recombination from states with a nonzero center-of-mass momentum. We find a clear separation of the two contributions up to 50 K. At higher temperatures, the zero-phonon line becomes so broad that the phonon sideband is only visible as a low-energy shoulder. It becomes evident in an asymmetrically broadened exciton transition toward lower energies. The energetic separation of ∼35 meV between the zero-phonon line and the phonon sideband is governed by the energy of the involved optical phonons. In contrast, the weak contribution of phonon absorption leads to a negligible indirect photoluminescence on the high energy side of the zero-phonon line.
Optical phonons also govern the broadening of the zerophonon line for temperatures above 100 K, while at lower temperature acoustic phonons dominate the PL line width, as will be further discussed later. The overall red-shift toward lower energies with decreasing temperature stems from a shift in the relative position of the conduction and valence bands due to polaronic effects 50 as well as the temperature-dependent contraction of the lattice. This phenomenon is known as the Varshni shift and can be described with the empirical formula 51 , where E g (0) is the band gap energy at The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter T = 0 K and α 1 , α 2 are material-specific constants. The red-shift is treated phenomenologically in this work, and the materialspecific constants are fitted to experimental data with E g (0) = 2.565 eV.
To be able to directly compare the theoretical prediction to experiment, we have also performed temperature-dependent PL measurements of the same 2D perovskites. The experimental PL spectra were collected from a monocrystalline (PEA) 2 PbI 4 thin sheet (∼100 nm) exfoliated from chemically synthesized crystals. To provide better stability of the exfoliated perovskite layer, it was encapsulated by h-BN sheets (for more details see ref 21). For temperature-dependent PL measurements, the sample was mounted on the coldfinger of a helium flow cryostat with a quartz optical window. PL was excited with the frequency-doubled output of a mode-locked Ti-sapphire laser and tuned to 400 nm. The excitation laser beam was focused on the sample using a 50× microscope objective with a numerical aperture of 0.55, giving a spot size of ∼1 μm diameter. The emitted PL was collected through the same objective and directed to a spectrometer equipped with a liquid nitrogen cooled charge-coupled device camera.
Figures 3e and 3f present a comparison of theoretically predicted and experimentally measured temperature-and energy-dependent PL spectra. We find a good qualitative agreement in terms of the appearance of the phonon sideband at the lower energy side of the zero-phonon line (cf. the rectangular boxes). In both experiment and theory, we find a pronounced phonon sideband appearing at 2.33 eV and staying clearly visible and distinguishable from the zero-phonon line for temperatures up to 50 K. Above this temperature the phonon sideband visible in experiment seems to merge/vanish with the main peak in contrast to theoretical predictions where this occurs around 100 K. We expect that this discrepancy can be related to h-BN encapsulation which might modify the motion of the inorganic layer and the related phonon mode with respect to the bulk case . 25 Indeed, in the case of the bulk (PEA) 2 PbI 4 crystals the phonon sideband can be resolved up to around 100 K . 20 This observation may indicate that the electron−phonon interaction can be substantially modified in thin layers of 2D perovskites due to possible residual strain induced by the organic cation interacting with the encapsulating layer . 52 It is also worth to note that in encapsulated layers the LO phonon energy is slightly lower than in bulk samples 20,25,36 which also points to the non-negligible impact of encapsulation.
The spectral broadening of excitonic resonances with temperature plays an important role for the visibility of phonon sidebands. Thus, we study the exciton−phonon scattering channels determining the temperature-dependent spectral line width of the three energetically lowest excitonic transitions. The line width is determined on a microscopic footing within the second-order Born−Markov approximation resulting in the momentum-dependent scattering rates given in eq 2. The appearing sum over all phonon modes and momenta includes all scattering processes that fulfill the energy conservation, that is, , where ℏΩ j q is the energy of the phonon with the mode j and momentum q, while ν E k and μ + E k q are the energies of the initial and final excitonic states. We include absorption (−) and emission (+) processes with optical and acoustic phonons. Because only the states within the light cone (i.e., k ≈ 0) contribute to the direct PL, the line width of the zero-phonon line is determined by the scattering rate Γ ν ≈ k 0 (cf. Figure 1b). The resulting line widths and their underlying microscopic scattering channels are displayed in Figure 4. We find spectral line widths of approximately 50, 25, and 15 meV for 1s, 2s, and 3s transitions at room temperature, respectively. In the case of 1s, we show that the scattering with acoustic phonons dominates the line width up to temperatures of ∼150 K (cf. the orange shaded region in Figure 4a). We find a linear dependence on temperature reflecting the behavior of the phonon occupations that are described by the Bose−Einstein distribution. Optical phonons start to be important for temperatures above 100 K (red region) and become the main channel governing the line width of 1s from ∼150 K. Optical phonons are important only at higher temperatures, since their energy is 35 meV, and thus the phonon occupation is negligibly small at low temperatures. There is also a temperature-independent contribution stemming from radiative recombination (gray region, 2.66 meV) that also limits the lifetime of 1s excitons in the light cone. The oscillator strength considerably decreases for higher transitions due to more spatially extended wave functions. Hence, the radiative The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter recombination has only a marginal contribution to their line widths with 1.9 × 10 −1 meV for 2s and 4.9 × 10 −2 meV for 3s (cf. Figure 4b,c). Note that we assume an ideal defect-free sample by disregarding the constant broadening Γ 0 that arises from scattering with defects and does not have any impact on the investigated temperature dependence. The quasi-linear dependence of the line width for higher states (2s and 3s) is caused by the fact that scattering with acoustic phonons is the main mechanism limiting the lifetime of 2s and 3s states at all temperatures. To explain the overall low importance of optical phonons for higher excitonic states, we investigate the wave function overlap ϕ ϕ ∑ α ν α μ * + k k k q , of the involved initial and final scattering states that enters in the exciton−phonon coupling elements. In particular, it is important to consider the specific momenta where energy conservation is fulfilled (cf. circular and triangular markers in Figure 5 denoting these momenta for optical and acoustic phonons, respectively). We find that the only possible scattering channel fulfilling the energy conservation for the 1s exciton is the intraband 1s → 1s scattering with optical and acoustic phonons, as shown in Figure 4d.
For the higher 2s and 3s states, multiple scattering channels are possible (cf. Figure 5b,c). Nevertheless, in all three cases it is the intraband scattering with acoustic phonons that shows the maximum wave function overlap due to the small momentum transfer of this transition. To be able to explain which specific scattering channel and phonon mode dominate the line width, it is not always sufficient to only consider the wave function overlap. The exciton−phonon coupling element is determined by the wave function overlap weighted by the corresponding deformation potential that is constant for optical phonons and linear in q for acoustic phonons. Furthermore, the temperature-and energy-dependent occupation of the involved phonon modes also needs to be considered since it directly enters the scattering rate (cf. eq 2). The combination of a strong electron−phonon coupling, large wave function overlap, and a moderate phonon number results in a very efficient intraband scattering with acoustic phonons that dominates the line width. The efficiency of scattering with optical phonons decreases for higher states despite the enhanced number of possible scattering channels as already seen in transition metal dichalcogenides. 45 This can be traced back to lower excitonic binding energies and thus more spatially extended excitonic wave functions. Because scattering with optical phonons provides a higher energy and momentum exchange than with acoustic phonons, the scattering strength will be determined by the overlap of two wave functions with significantly different momenta. Because the wave functions of higher excitonic states are more localized in momentum space, there is only a small overlap of initial and final state wave functions giving rise to a large suppression of scattering processes with optical phonons.
While acoustic phonons predominantly induce scattering within an excitonic band, optical phonons can also bridge larger energy distances between different bands. To better understand these processes, we further decompose the most important scattering channels with optical phonons in Figure  4d−f. Because for optical phonons the coupling strength and phonon number do not depend on momentum, the efficiency of a particular scattering channel is only determined by the wave function overlap. We find that the most important contribution caused by optical phonons corresponds to emission processes to the energetically lower 1s state (blue areas in Figure 4e,f). Note that spontaneous emission is much more likely than stimulated emission at low temperatures, where the phonon number is very small. Moreover, the fact that spontaneous emission does not depend on the phonon number results in the constant line width at low temperatures. For 2s excitons, the wave function overlap for the optical phonon absorption processes 2s → 3s and 2s → 3d stands out  . Wave function overlap of the excitonic states involved in phonon-driven transitions, which determine the spectral line width of (a) 1s, (b) 2s, and (c) 3s states. The circular (triangular) markers indicate the momentum at which the scattering can take place via optical (acoustic) phonons. Here, the energy conservation is fulfilled. Note that the strongest wave function overlap involving the transitions within the same band have been multiplied by a factor of 0.5 or 0.3 to be able to better illustrate other scattering channels.
The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter (cf. Figure 5b), which is directly reflected in their important contributions to the optical phonon portion of the line width in Figure 4e. The 3s exciton state is close to the continuum allowing multiple scattering channels driven by emission and absorption of phonons. The scattering with optical phonons at the 3s state is dominated by the 3s → 1s transition induced by spontaneous emission (cf. Figure 4f). Furthermore, because of the energetic proximity of other states, significant wave function overlap exists for interband acoustic phonon scattering processes (cf. Figure 5c). However, the small phonon number at the large momenta involved in these scattering processes drastically reduces the efficiency of these channels, resulting in an even smaller line width at room temperature compared to the 1s exciton exhibiting much fewer scattering channels.
In summary, we have presented a theoretical study supported by experimental measurements shining light on excitonic properties of monolayered halide perovskites. On the basis of a quantum mechanical approach, we have determined the eigenfunctions and binding energies of the entire Rydberglike series of excitonic states. Furthermore, we have calculated and measured temperature-dependent photoluminescence spectra with a particular focus on the emergence of lowenergy phonon sidebands. Finally, we have provided microscopic insights into the exciton−phonon scattering channels governing the temperature-dependent spectral line width of the three energetically lowest excitonic transitions. Overall, our work contributes to a better understanding of the optical properties of 2D perovskites and can guide future studies in this growing field of research.