We report the low-frequency Raman spectrum (ω = 10 cm−1–150 cm−1) of a wide variety of alkylammonium iodide based 2D lead halide perovskites (2D LHPs) as a function of A-site cation (MA = methylammonium and FA = formamidinium), octahedral layer thickness (n = 2–4), organic spacer chain length (butyl-, pentyl-, hexyl-), and sample temperature (T = 77 K–293 K). Using density functional theory calculations under the harmonic approximation for n = 2 BA:MAPbI, we assign several longitudinal/transverse optical phonon modes between 30 cm−1 and 100 cm−1, the eigendisplacements of which are analogous to that observed previously for octahedral twists/distortions in bulk MAPbI. Additionally, we propose an alternative assignment for low-frequency modes below this band (<30 cm−1) as zone-folded longitudinal acoustic phonons corresponding to the periodicity of the entire layered structure. We compare measured spectra to predictions of the Rytov elastic continuum model for zone-folded dispersion in layered structures. Our results are consistent across the various 2D LHPs studied herein, with energetic shifts of optical phonons corresponding to microscopic structural differences between materials and energetic shifts of acoustic phonons according to changes in the periodicity and elastic properties of the perovskite/organic subphases. This study highlights the importance of both the local atomic order and the superlattice structure on the vibrational properties of layered 2D materials.

Over the past decade, hybrid organic–inorganic perovskites have emerged as candidate materials for a variety of applications such as efficiency-competitive solar cells,1–5 color tunable light emitting diodes (LEDs),6,7 and sensitive photodetectors.8 This interest stems from a range of physical properties such as high optical absorption coefficients, small exciton binding energies, and long/balanced electron–hole diffusion lengths.9,10 These materials are denoted via the chemical formula ABX3 and invariably exhibit a crystal structure in which a small organic cation (A), such as methylammonium (MA) or formamidinium (FA), is enclosed within corner-sharing metal (B) halide (X) octahedra. The structural, optical, and electronic properties of these hybrid perovskites can be carefully engineered via, among other techniques, synthetic control of the composition of these constituents.

Despite the exceptional qualities espoused above, devices utilizing hybrid perovskites such as MAPbI3 have been diminished by insufficient long-term stability that stems primarily from the complexation of water in ambient moisture with the A-site organic cation.11,12 One solution to this problem is to stabilize atomically thin sheets of perovskite materials within an insulating secondary organic phase consisting of larger molecules than would fit within a traditional perovskite lattice.13–20 This solution utilizes a nanostructured 2D layered architecture to effectively protect the A-site cations from external moisture. Furthermore, such 2D lead halide perovskites (2D LHPs) exhibit blue-shifted emission spectra due to the formation of strongly bound excitonic states within the 2D octahedral layers, which can be advantageous for some applications.21–23 2D LHPs have already begun to show promising performance metrics when used in solar cells,16,24–28 LEDs,29,30 and photodetectors.31,32

The low-frequency vibrational spectrum of these materials, which informs on the phonons supported by the crystal, provides a useful fundamental insight into a variety of important phenomena relevant for device engineering. For example, with respect to bulk MAPbI3, Raman spectroscopy is utilized to help identify localized crystallographic transformations or degradation pathways, particularly those that couple with the electronic behavior of the material.33 Additionally, this technique provides critical corroborating details for the analysis of various particle-phonon scattering processes (e.g., electron–phonon, phonon–phonon, and spin-phonon), such as the energies and lifetimes of relevant optically active phonon modes.34,35 These mechanisms are essential in developing quantitative models for energy transport in devices that incorporate these materials. This utility can be extended to include experimental verification of theoretical calculations of important relationships such as the phonon dispersion and vibrational density of states, typically through techniques such as density functional theory (DFT) or molecular dynamics simulation.36–40 Finally, for hybrid organic–inorganic materials, wherein phases of incongruous elastic properties commingle, Raman spectroscopy can help elucidate the vibrational overlap between the constituent phases. Due to the heavy atomic masses of Pb and I, phonon modes in hybrid lead iodide perovskites are observed at lower frequencies (<100 cm−1) than traditional all-inorganic semiconductors.

For bulk 3D lead halide perovskites, the observed low-frequency Raman spectrum has been found to be particularly sensitive to the crystal structure of the material.36,38,40–42 In particular, for higher temperature phases in which the translational and rotational freedom of the A-site cation is unlocked, a broad central (0 cm−1) peak is observed and ascribed to the local polar fluctuations of this cation.38 At lower temperatures, particularly in the orthorhombic crystal phase, for which the cation is largely immobilized, well-resolved features are obtained although in many cases these features remain dynamically broadened by local inhomogeneities in cation positioning. Nonetheless, density functional theory calculations verify that the low-frequency phonon modes arise due to both the lead halide cage modes (octahedral distortions) and associated coupled movement of the organic cation.36,39,41–44

Layered perovskite-type compounds (sometimes called low-dimensional networked perovskites or n = 1 2D LHPs), a subset of the 2D LHP family in which single atomic layers of BX6 octahedra are separated by an organic sublayer, have also been investigated with Raman spectroscopy.45–51 Like their bulk 3D counterparts, these 2D layered compounds exhibit low-frequency Raman-active phonon modes that are attributed to octahedral twists and distortions, which are coupled to the motions of the molecules within the organic subphase.49 These modes, as for bulk perovskites, are also dynamically broadened by molecular motion of the organic subphase.

In this article, we focus specifically on thicker members (n = 2–4) of the iodide 2D LHP family. We report the low-frequency Raman spectrum (ω = 10 cm−1–150 cm−1) as a function of A-site cation (MA, FA), octahedral layer thickness (n = 2–4), organic spacer chain length (butyl-, pentyl-, hexyl-), and sample temperature (T = 77 K–293 K). We use density functional theory (DFT) calculations under the harmonic approximation for n = 2 BA:MAPbI to assign several longitudinal/transverse optical (LO/TO) phonon modes between 30 cm−1 and 100 cm−1. For these modes, we find that the eigendisplacements are similar to those previously observed for octahedral twists/distortions in bulk MAPbI. We also propose an alternative assignment of the low-frequency modes below this band (<30 cm−1) as zone-folded longitudinal acoustic (zf-LA) phonons. These modes represent coherent propagating acoustic waves attributed to the periodicity of the entire layered structure. We support this assignment in the context of all-inorganic superlattices using the Rytov elastic continuum model for zone-folded dispersion in layered structures. We apply our assignment across the various 2D LHPs studied herein and observe energetic shifts of optical phonons corresponding to microscopic structural differences between materials and energetic shifts of acoustic phonons according to changes in the periodicity and elastic properties of the perovskite/organic subphases. This study underlines not only the importance of the local atomic order but also that of the superlattice structure/composition in influencing the vibrational properties of layered 2D materials.

2D lead halide perovskite (2D LHP) crystals were made via a cooling-induced crystallization method reported in detail elsewhere.52 A large volume of lead (II) iodide (PbI2) was prepared by dissolving lead (II) oxide (PbO) in a hydriodic acid (HI) solution under reflux at 130 °C. After 15 min, the solution was then cooled down to room temperature before a small volume of it was transferred into a vial containing hypophosphorus acid (H3PO2). H3PO2 acts as a strong reducing agent, converting any residual I2 to HI, yielding a bright yellow solution. The long-chain aklyamine (butylamine [BA], pentylamine [PA], or hexylamine [HA]), which will ultimately form the organic spacer layer within the 2D LHP, was then added dropwise into the acidic solution in an ice bath, forming an orange precipitate of n = 1 (BA,PA,HA)2PbI4 2D LHP. In a separate vial, methylammonium iodide (or formamidinium iodide) salt was dissolved in an HI solution and subsequently added to the solution containing the n = 1 precipitate. The combined solution was heated on a hotplate at 130 °C for 4 min, until the solution turned clear. Crystals were then grown by storing the clear solution in a thermos filled with hot sand at 110 °C overnight. Crystals were isolated by suction filtration and dried under reduced pressure for at least 12 h.

Raman spectra were collected in a backscattered geometry using a home-built micro-Raman instrument. Samples were housed within an optical cryostat (Janis ST-500, fused quartz window) mounted on an inverted Nikon microscope (60×, 0.6 NA objective) and kept under vacuum during all measurements. A 785 nm narrow-band CW excitation source was filtered from undesirable amplified spontaneous emission using a series of cleanup filters (laser and filters from Ondax). The Rayleigh line was minimized by passing the collected signal through a set of volume holographic grating notch filters (from Ondax) before being dispersed in a 0.5 m focal length spectrograph (SP-2500, Princeton Instruments) using a 1200 g/mm and 750 nm blaze grating and ultimately imaged on a cooled CCD camera (Princeton Instruments Pixis). The Rayleigh notch filters, centered at 0 cm−1, have a full attenuation bandwidth of ±10 cm−1. The overall spectral resolution of the instrument is 0.9 cm−1. The collected spectra were fit using the multipeak fitting package in Igor Pro 6.37. A sloping Gaussian background was used to approximate the tails of the broad central scattering feature beyond ω = ±10 cm−1, and Gaussian line shapes were found to best fit the Raman features present in all samples. For data collected at different temperatures, the fit from 77 K was used as an initial guess for subsequent fits at higher temperatures (see Fig. S2 for the representative data sample). Data were collected at several locations along a pristine crystal to verify reproducibility of the measured spectra. Errors reported represent the 99% confidence interval for the fit parameters.

All electronic structure calculations were carried out using DFT,53,54 as implemented in the Vienna Ab Initio Simulation Package (VASP v5.4),55,56 integrated in the MedeA® computational environment.57 Exchange-correlation effects were modeled using the generalized gradient approximation (GGA) through the Perdew–Burke–Ernzerhof (PBE)58 functional. The Kohn–Sham equations were solved with the frozen-core projector augmented wave (PAW)59 method, using plane-wave basis sets with a kinetic energy cutoff of 800 eV. In addition, the D3 correction scheme60 was employed to improve the description of London dispersion interactions. A Γ-centered, 3 × 3 × 2 Monkhorst–Pack k mesh was used for all simulations. All initial geometries were optimized for zero external pressure, and atomic forces no greater than 1 meV/Å.

Lattice dynamics simulations were performed in the framework of the direct method,61 also implemented in the MedeA software. Harmonic force constants were computed using small displacements around the set of symmetry-reduced atomic equilibrium positions, considering an interaction range of about 10 Å. The non-analytic contribution to the dynamical matrix was accounted for by calculating the Born effective charge tensors, leading to the splitting of LO and TO modes at q → 0. The relative contribution fZω of element Z to a certain vibrational mode ω was evaluated as

fZω=i=Zϵiω2iϵiω2,

where ϵiω is the eigenvector of mode ω associated with the ith atom.

Finally, the Raman activity tensors were determined by evaluating the derivative of the polarizability tensor α with respect to the normal modes previously calculated.62 That is, for a set of small atomic displacements, we have calculated the static dielectric matrix using density functional perturbation theory, including self-consistent field effects to the first order.63 Scalar Raman intensities corresponding to unpolarized light and polycrystalline samples were obtained by considering a range of scattering parameters in the expression of the first-order differential Raman cross section for Stokes processes.62 

The measured Raman spectrum of n = 2 BA:MAPbI (BA = butylammonium and MA = methylammonium) perovskite crystals is shown in Fig. 1(b). At room temperature, the observed spectrum is dominated by a broad (±50 cm−1) symmetric scattering feature centered at 0 cm−1 (the central ±10 cm−1 portion of this scattering feature is attenuated by the notch filter). Analogous to the higher temperature Raman spectra of the bulk MAPbI3 perovskite, this broad central feature likely arises due to the presence of local polar fluctuations of the organic cations (both the A-site cations and the longer chain spacers). Specifically, interactions between the cations and the inorganic lattice, due to translation/rotation of the former, over a range of time scales (∼0.3 ps to 3 ps for MA cations in MAPbI3) fast relative to the lifetimes of the Raman-active phonons lead to a broad peak in the frequency domain centered at 0 cm−1.38,64 While it is not immediately discernable which cation contributes more to the central peak observed, it is well established that at room temperature, both components exhibit a significant dynamic disorder.52,65 Although the crystal structure of the 2D LHP is orthorhombic owing to the larger periodicity inherent in the material, the inorganic sub-lattice closely approximates that of cubic MAPbI3, leaving a cavity large enough for the MA cation to rotate.52 Additionally, both the x-ray diffraction measurements and thermal analysis indicate that the BA molecules in the organic subphase behave, analogous to self-assembled monolayers, as dynamically disordered molecules bound to a two dimensional surface.66 Although all Raman features virtually disappear beneath this broad central scattering envelope at room temperature, one mode at ∼44 cm−1 remains visible.

FIG. 1.

(a) Space filling model extracted from single crystal x-ray diffraction for a triclinic crystal structure of n = 2 BA:MAPbI and phase bar showing structural transitions and approximate transition temperatures. The red dotted line traces a single unit cell. (b) Low-frequency Raman spectrum at room temperature (red) and 77 K (blue). Raman frequencies and activities predicted from DFT calculations (green). (c) Calculated dispersion for the folded Brillouin zone from the Rytov model (blue) with experimentally measured mode frequency overlaid for clarity (red line).

FIG. 1.

(a) Space filling model extracted from single crystal x-ray diffraction for a triclinic crystal structure of n = 2 BA:MAPbI and phase bar showing structural transitions and approximate transition temperatures. The red dotted line traces a single unit cell. (b) Low-frequency Raman spectrum at room temperature (red) and 77 K (blue). Raman frequencies and activities predicted from DFT calculations (green). (c) Calculated dispersion for the folded Brillouin zone from the Rytov model (blue) with experimentally measured mode frequency overlaid for clarity (red line).

Close modal

To reduce the scattering strength of the broad central feature and reveal the underlying low-frequency vibrational spectrum of the ordered lattice, data were collected at 77 K, which is far below the freezing point of the organic subphase. At these temperatures, the crystal structure differs somewhat from that at room temperature. The 2D LHP as a whole is characterized by a lower symmetry triclinic structure [Fig. 1(a)], owing to the contraction of the organic subphase during freezing and a distortion of the inorganic sub-lattice in response. At this temperature, several phonon modes between 20 cm−1 and 100 cm−1 become visible. In addition to these modes, however, the MA cation still exhibits some dynamic disorder, and thus, the central peak is still present in this spectrum.

First principles lattice-dynamics calculations were performed under the harmonic approximation, using the experimentally determined triclinic crystal structure as an input, to determine the vibrational modes of the material, the Raman intensity corresponding to each mode, and the real-space vector displacements within the crystal corresponding to each mode (see the section titled “Methods” and Fig. S1). The brightest Raman-active modes determined in this fashion were then utilized to assign the experimentally obtained spectrum at 77 K. As the calculations were performed at 0 K under the harmonic approximation, small shifts (of about 4 cm−1–5 cm−1) between the calculated and experimental frequencies are expected. The harmonic approximation does not consider anharmonic bond expansion at non-zero temperatures, which leads to the said shifts.

We separate our discussion of the observed/calculated modes into three distinct bands: those above 60 cm−1, those between 30 cm−1 and 60 cm−1, and those below 30 cm−1. Above 60 cm−1, one distinct mode is observed at 100 ± 1 cm−1. While this frequency is likely still too low for the observation of isolated vibrations within the aliphatic chains of the organic subphase, the coupling of molecular modes to inorganic vibrations in bulk MAPbI3 leads to Raman-active vibrations in this region.36 Density functional theory calculations of n = 2 BA:MAPbI suggest that this is indeed the case for these layered crystals as well. The brightest Raman features calculated (at 75 cm−1 and 111 cm−1) in this region correspond to TO modes in which the organic and inorganic sublattices exhibit in-plane shearing [Fig. 2(a)]. Experimentally, this peak appears with relatively low intensity, in agreement with the low calculated activity of the mode, and large FWHM as a result of inhomogeneous broadening with respect to the organic cation.

FIG. 2.

Real-space eigendisplacements determined via DFT for the brightest Raman features in the (a) “high” frequency range (>60 cm−1), (b) mid-frequencies (30 cm−1–60 cm−1), and (c) lowest frequencies (<30 cm−1). The bars to the left of each space filling model represent the elementwise eigenfractions, highlighting the dominant contributors to each mode.

FIG. 2.

Real-space eigendisplacements determined via DFT for the brightest Raman features in the (a) “high” frequency range (>60 cm−1), (b) mid-frequencies (30 cm−1–60 cm−1), and (c) lowest frequencies (<30 cm−1). The bars to the left of each space filling model represent the elementwise eigenfractions, highlighting the dominant contributors to each mode.

Close modal

Between 30 cm−1 and 60 cm−1, several Raman-active modes corresponding to octahedral twists/distortions are observed experimentally. In particular, a sharp mode at 47.5 ± 0.1 cm−1 can be clearly resolved in addition to several overlapping peaks between 30 cm−1 and 40 cm−1. DFT calculations suggest that these modes primarily correspond to distortions of the inorganic cage, with minor associated displacement within the organic subphase [Fig. 2(b)]. While these modes are appreciably sharper than those observed at higher frequency, they are, nonetheless, well fit by Gaussian line shapes, suggesting that significant inhomogeneous broadening remains. This broadening could be the result of either strain-induced deformation of the inorganic lattice or a distribution of equilibrium positions with respect to the MA cation, both of which are observed via x-ray diffraction measurements.52 

Below 30 cm−1, one prominent Raman-active vibrational mode is observed experimentally for the n = 2 BA:MAPbI 2D LHPs (at 25.2 cm−1). Interestingly, there are no corresponding bright Raman modes calculated from DFT. The energy of this mode is comparable to that observed for an octahedral twist (TO) mode in bulk cubic MAPbI3, but the equivalent vibrational mode is not observed with significant Raman activity in our calculations. Our calculations do, however, reveal one mode in the vicinity of that measured experimentally (at 29 cm−1), but with much lower intensity than observed. (We note that the DFT Raman scattering intensities are simulated by averaging over many different crystallographic orientations, whereas experimental Raman spectra were collected by backscattering from 2D perovskite crystals whose constituent layers were most likely oriented perpendicular to the incident laser beam.) The calculated mode corresponds to a distortion of the inorganic sublattice, specifically a contraction in-plane in conjunction with an out-of-plane expansion [Fig. 2(c)]. We suggest that if this calculated mode contributes to the peak observed experimentally, then it does so at a comparable intensity to the peak observed at 100 cm−1. Such a peak would likely not be distinguishable atop the growing background produced by the central peak at low wavenumbers; hence, we suggest that an alternative vibrational mode may be responsible for this feature.

While density functional theory calculations provide a deep insight into the phonon dispersion within the first Brillouin zone as determined by the crystallographic unit cell, it cannot provide insights into the effect of periodicities comparable to or longer than this. For most 3D crystals, this is sufficient to account for all relevant phonon modes, but for periodic nanostructures such as 2D all-inorganic superlattices, this difference is critical in understanding the low-frequency phonon dispersion (and the resulting Raman spectrum).

The longer cross-plane periodicity (d) introduced by the stacking of materials with differing elastic properties atop one another can lead to phonon dispersion corresponding to the “mini” Brillouin zone (with wavevector less than π/d). Within this mini-Brillouin zone, acoustic phonon branches are “zone-folded” and even form mini gaps in their dispersion for wave vectors that satisfy the Bragg condition. This phenomenon is analogous to the formation of electronic bandgaps for an electron in a periodic potential well, as described by the Kronig–Penney model. In the case of zone-folded longitudinal acoustic (zf-LA) modes, the physical picture is one of the collective longitudinal compressions and expansions that coherently propagate throughout the multilayered stack of material. Additionally, because zf-LA modes have non-zero frequency at the zone center, they are optically active and can be probed using Raman spectroscopy in a backscattering geometry. Such phonons are well established for various all-inorganic superlattices67–71 and have even been observed in some organic–inorganic layered materials.72 

When the interface between layers in a repeating bilayer superlattice is sharp and the dispersion can be approximated as linear near the zone center, the longitudinal zone-folded acoustic modes can be represented with a simple continuum model, first given by Rytov,73 

cos(kd)=cosωd1υ1cosωd2υ21+κ22κsinωd1υ1sinωd2υ2,κ=ρ1υ1ρ2υ2,
(1)

where k is the wavevector, ω is the frequency, v1,2 are sound velocities within the two phases, d1,2 are the respective layer thicknesses, and ρ1,2 are mass densities within each phase.

In many ways, 2D LHPs exemplify an ideal superlattice as they consist of atomically abrupt interfaces, virtually no crystalline heterogeneities (as evidenced by their well-established octahedral thickness purity), and very long-range order stemming from their growth as single crystals.65 Importantly, however, 2D LHPs involve a significant mismatch in the elastic properties of the organic and inorganic subphases. Coherent longitudinal acoustic phonons near 0 cm−1 (optically inactive) have actually been observed for this variant of 2D LHPs using stimulated Brillouin scattering although whether or not these materials exhibit zone-folding has not been discussed.74 As a result, we suggest the use of the Rytov model for phonon dispersion within the mini-Brillouin zone to describe the longitudinal acoustic (LA) phonons inherent to 2D LHPs.

Using bulk values for the sound velocity and density in the inorganic sub-lattice and those measured for SAMs and bulk liquids for the organic subphase and using layer thicknesses calculated from single crystal x-ray diffraction data, we calculate the zone-folded phonon dispersion for n = 2 BA:MAPbI [Fig. 1(c), Table I]. The use of bulk values to approximate the properties of the inorganic subphase carries with it an assumption that should be emphasized. Specifically, this usage assumes that the perovskite subphase approximately retains its bulk elastic properties even at the nanometer-scale period thicknesses exhibited by these materials. The applicability of this approximation is apparent for the mass density based on previously reported x-ray diffraction measurements,52 but may be less clear with regard to the sound velocity. It has been reported through multiple experimental and computational studies that the mean free path of phonons in bulk perovskites is on the length scale of a single unit cell,75–77 implying that the bulk sound speed should be maintained even down to nanometer thicknesses. That stated, this approximation remains as a notable simplifying assumption made in utilizing the Rytov model.

TABLE I.

Summary of input parameters used for the Rytov Model.a

n = 2 BA:MAPbIn = 2 BA:FAPbIn = 3 BA:MAPbIn = 4 BA:MAPbIBAPAHA
d (Å) 12.6 12.8 18.9 25.2 8.1 9.9 
n = 2 BA:MAPbIn = 2 BA:FAPbIn = 3 BA:MAPbIn = 4 BA:MAPbIBAPAHA
d (Å) 12.6 12.8 18.9 25.2 8.1 9.9 
Density (g/cm3)Sound velocity (m/s)
MAPbI 4.180  139080  
FAPbI 4.181  127078  
Butyl- 0.57b 40082c 
Pentyl- 0.63b 70082c 
Hexyl- 0.69b 90082c 
Density (g/cm3)Sound velocity (m/s)
MAPbI 4.180  139080  
FAPbI 4.181  127078  
Butyl- 0.57b 40082c 
Pentyl- 0.63b 70082c 
Hexyl- 0.69b 90082c 
a

Period thicknesses were extracted directly from previously reported single crystal x-ray diffraction of identical 2D LHPs.52 

b

Standard densities of liquid alkanes were used to approximate the density of the organic subphase.

c

Data published in Ref. 82 were linearly extrapolated to estimate the sound velocities for bound alkyl groups studied herein.

Unsurprisingly—owing to the low sound velocities of both subphases—the Rytov zone-folded phonon dispersion is characterized by largely dispersionless branches. Additionally, the large mismatch between the acoustic impedances of the organic and inorganic subphases (directly calculated via κ as ∼25.1) opens the calculated bandgaps significantly wider than those calculated for all-inorganic superlattices.68 This calculation reveals an optically active zone-folded LA phonon mode at 25.2 cm−1, in excellent agreement with that observed experimentally. Thus, we assign the prominent peak at 25.2 cm−1 as a zone-folded LA phonon of the superlattice. Importantly, we do not observe any higher frequency zone-folded modes in our spectra because those modes would likely extend beyond the first Brillouin zone of the 2D LHP. This is because, broadly, there can be no zone-folded LA modes at frequencies that would correspond to a wavelength shorter than the period thickness of the 2D LHP. Additionally, the intensity of any higher frequency modes would be of progressively lower intensity and would be unlikely to be distinguished from the optical modes. Additionally, although the Rytov dispersion suggests further zone-folded LA modes at lower frequencies, these modes are likely hidden by a combination of the optical filter used to suppress the Rayleigh line (onset at ∼20 cm−1) and the increasingly high intensity contribution of the central peak at lower wavenumbers. As we will show briefly, for 2D LHPs with a less prominent central scattering feature, we can resolve even these zone-folded LA phonons.

Altering the A-site cation: Assignment of Raman modes in n = 2 BA:FAPbI

To further investigate the nature of the modes assigned above, we vary the identity of the A-site cation in the 2D LHP from methylammonium (MA) to the larger formamidinium (FA) cation. As for bulk lead halide perovskites, this substitution does not appreciably change the crystal symmetry. At room temperature, the organic subphase is disordered and the space group is orthorhombic. Similarly, below the 2D melting point of the organic subphase, the symmetry is reduced to a triclinic P-1 cell, with ordered and oriented organic molecular chains [Fig. 3(a)]. One key difference between the 2D LHPs arises from the relative stiffness of the FAPbI inorganic sublattice as compared to that of MAPbI. Perovskites utilizing the large FA cation in the A-site are appreciably less stiff (lower elastic modulus) than analogous perovskites with smaller central cations.78 This is likely because the large FA cation, which is essentially as large as a cation can be sustained by the octahedral network (quantified by a near unity Goldschmidt tolerance factor), stretches the size of the unit cell. The longer bond lengths correspond to weaker elastic stiffness and consequently a lower sound velocity for effectively the same material density.78,79

FIG. 3.

(a) Space filling model extracted from single crystal x-ray diffraction for the triclinic crystal structure of n = 2 BA:FAPbI and phase bar showing structural transitions and approximate transition temperatures. (b) Low-frequency Raman spectrum at room temperature (red) and 90 K (blue). (c) Calculated dispersion for the folded Brillouin zone from the Rytov model (blue) with experimentally measured mode frequencies overlaid for clarity (red line/shaded area).

FIG. 3.

(a) Space filling model extracted from single crystal x-ray diffraction for the triclinic crystal structure of n = 2 BA:FAPbI and phase bar showing structural transitions and approximate transition temperatures. (b) Low-frequency Raman spectrum at room temperature (red) and 90 K (blue). (c) Calculated dispersion for the folded Brillouin zone from the Rytov model (blue) with experimentally measured mode frequencies overlaid for clarity (red line/shaded area).

Close modal

Owing to their structural similarity, the room temperature low-frequency Raman spectrum of n = 2 BA:FAPbI is almost identical to that acquired for n = 2 BA:MAPbI [Fig. 3(b)]. At cryogenic temperatures, however, the observed spectrum differs. Analogous peaks above 25 cm−1 are observed and likely correspond to the same eigendisplacements calculated for the n = 2 BA:MAPbI 2D LHP above. The zone-folded LA mode, however, actually redshifts slightly to 24.8 ± 0.2 cm−1 from that measured for n = 2 BA:MAPbI. This slight shift intuitively stems from the lower sound velocity of the FAPbI sub-lattice, a relationship that is well represented by the Rytov model, which predicts a decrease in the zone-folded LA mode by ∼0.1 cm−1 [Fig. 3(c)]. The characteristic frequency of the inorganic medium, given by v1/d1 in Eq. (1), is directly proportional to the sound velocity of the material for roughly equivalent slab thicknesses. Additionally, and somewhat surprisingly, several lower frequency phonon modes are observed between 10 cm−1 and 25 cm−1 (a sharp peak at ∼10 cm−1 and a broader band between 15 cm−1 and 20 cm−1). The phonon energy of these modes is all roughly consistent with the phonon dispersion given by the Rytov model for n = 2 BA:FAPbI; thus, we tentatively assign all these modes as zone-folded LA phonons as well. This agreement over several different branches of the Rytov dispersion further supports our assignment of these modes. We attribute the appearance of these additional peaks in the FA spectrum to a decrease in the dynamic disorder exhibited by the larger, likely sterically limited, FA cation. This consequentially decreases the intensity of the central broad scattering peak below 20 cm−1 where the optical filter begins attenuating the Raman signal, allowing for the resolution of features that were previously unresolvable.

Increasing the octahedral thickness: Assignment of Raman modes in n = 3,4 BA:MAPbI

Increasing the PbI octahedral layer thickness, n, does not significantly impact the overall symmetry of the 2D LHP. Both n = 3 and n = 4 2D LHPs retain orthorhombic space groups at high temperature and low symmetry triclinic cells below the freezing point of the organic subphase [Fig. 4(a)]. Consequently, at all temperatures and n-values, the Raman spectra remain qualitatively the same [Figs. 4(b) and 4(c)]. One subtle distinction arises from the presence of PbI octahedra with differing local symmetry for higher values of n. Specifically, although for n = 2 all octahedra interact with the ammonium ions from both the organic spacer and MA, higher-n 2D LHPs have additional “internal” octahedra that only engage in hydrogen bonding with MA [(colored purple in Fig. 4(a)]. For these internal octahedra, the local environment is identical to that of bulk MAPbI3. External octahedra [colored green in Fig. 4(a)], on the other hand, are categorized by axial bond-length asymmetry, wherein the axial Pb–I bond lengths are distorted significantly from the values for bulk Pb–I.52 Since the optical phonons assigned for bulk MAPbI3 previously, as well as their analogs assigned for n = 2 BA:MAPbI above, are twists and distortions of presumably identical octahedra, it is expected that asymmetry therein would lead to slight changes in the phonon energies. Indeed, at cryogenic temperatures, we observe slight red shifting of the prominent optical cage mode as n is increased [Fig. 4(e)]. This is to say, as the relative quantity of internal octahedra is increased, the phonon energy shifts toward its bulk value (45 cm−1).36 

FIG. 4.

(a) Space filling model extracted from single crystal x-ray diffraction for the triclinic crystal structure of n = 4 BA:MAPbI and phase bar showing structural transitions and approximate transition temperatures for both n = 3 and n = 4 BA:MAPbI. External octahedra are highlighted in teal, while internal octahedra are highlighted in purple. The red dotted line traces a single unit cell. (b) Low-frequency Raman spectrum of n = 4 BA:MAPbI at room temperature (red) and 77 K (blue). (c) Low-frequency Raman spectrum of n = 3 BA:MAPbI at room temperature (red) and 77 K (blue) and (d) zone-folded (z-f) LA phonon energy as a function of octahedral layer thickness. (e) Optical phonon energy as a function of octahedral layer thickness. Bulk value is obtained from Ref. 31.

FIG. 4.

(a) Space filling model extracted from single crystal x-ray diffraction for the triclinic crystal structure of n = 4 BA:MAPbI and phase bar showing structural transitions and approximate transition temperatures for both n = 3 and n = 4 BA:MAPbI. External octahedra are highlighted in teal, while internal octahedra are highlighted in purple. The red dotted line traces a single unit cell. (b) Low-frequency Raman spectrum of n = 4 BA:MAPbI at room temperature (red) and 77 K (blue). (c) Low-frequency Raman spectrum of n = 3 BA:MAPbI at room temperature (red) and 77 K (blue) and (d) zone-folded (z-f) LA phonon energy as a function of octahedral layer thickness. (e) Optical phonon energy as a function of octahedral layer thickness. Bulk value is obtained from Ref. 31.

Close modal

Changing the inorganic slab thickness alters the periodicity and the characteristic frequency within the Rytov model and thus should lead to some variation of the zone-folded LA Raman features from those observed for n = 2. As for the n = 2 2D LHPs discussed above, higher-n 2D LHPs also appear to support such zone-folded LA modes although at increasing values of n, the higher frequency optical modes (such as that analogous to the bulk MAPbI3 observed near 26 cm−1) will soften and begin to overlap the LA phonons. At high enough n-values, the periodicity of the 2D LHP will become too large relative to the wavelength of the phonons, and LA phonons will cease to be observed. This, naturally, is another expression of the tendency toward bulk properties of a nanomaterial as its characteristic size is increased relative to the relevant characteristic length of confinement. The increase in period thickness for n = 3 and 4 BA:MAPbI leads to a reduction in the characteristic frequency, which is large enough to yield an increase in the number of branches within the mini-Brillouin zone (Fig. S9). This additional branch intersects with the zone center at values comparable to those observed experimentally [Fig. 4(d)]. Interestingly, for both n = 3 and n = 4, two branches intersect the zone center at frequencies within the linewidth of the feature we have assigned as a zone-folded acoustic phonon, implying that these features are potentially doublets.

Altering the organic subphase: Assignment of Raman modes in n = 2 (PA and HA):MAPbI

In addition to varying the perovskite sublattice, we also synthesized 2D LHPs insulated with various organic spacer molecules. Specifically, we varied the alkyl chain length of the organic subphase from a four-carbon chain (BA, above) to five- and six-carbon chain alkylammonium cations (PA = pentylammonium and HA = hexylammonium). At cryogenic temperatures (well below the freezing point of the hexyl and pentyl amine subphases), the crystal structures vary slightly from those observed for the 2D LHPs that utilize BA as an organic spacer [Figs. 5(a) and 5(c)]. Rather than a triclinic space group, the hexylammonium-spaced 2D LHPs are described by monoclinic unit cells at 77 K [Fig. 5(c)]. Additionally, since these unit cells contain many more atoms than for triclinic n = 2 BA:MAPbI, it was infeasible to carry out DFT calculations for these materials. Since the optical modes we observe in our Raman measurements are primarily inorganic perovskite cage modes, however, and given the qualitative similarity between our results for all 2D LHPs, we hypothesize that the modes we observe are analogs of those assigned above for n = 2 BA:MAPbI. Interestingly, the relative intensity of the cage modes between 25 cm−1 and 45 cm−1 increases significantly with the symmetry of the crystal structure [Figs. 5(b) and 5(d)], a trend which qualitatively matches that observed for bulk lead halide perovskites via DFT calculations.36 This suggests that the increased symmetry leads to less splitting of phonon modes and thus higher Raman intensity.

FIG. 5.

(a) Space filling model extracted from single crystal x-ray diffraction for the triclinic crystal structure of n = 2 PA:MAPbI and phase bar showing structural transitions and approximate transition temperatures. (b) Low-frequency Raman spectrum at room temperature (red) and 77 K (blue). (c) Space filling model extracted from single crystal x-ray diffraction for the monoclinic crystal structure of n = 2 HA:MAPbI and phase bar showing temperature invariance of the crystal structure. (d) Low-frequency Raman spectrum at room temperature (red) and 77 K (blue).

FIG. 5.

(a) Space filling model extracted from single crystal x-ray diffraction for the triclinic crystal structure of n = 2 PA:MAPbI and phase bar showing structural transitions and approximate transition temperatures. (b) Low-frequency Raman spectrum at room temperature (red) and 77 K (blue). (c) Space filling model extracted from single crystal x-ray diffraction for the monoclinic crystal structure of n = 2 HA:MAPbI and phase bar showing temperature invariance of the crystal structure. (d) Low-frequency Raman spectrum at room temperature (red) and 77 K (blue).

Close modal

Altering the chemical composition of the organic subphase naturally impacts the zone-folded LA (zf-LA) phonons as well, which are observed for both PA and HA 2D LHPs studied herein. Changing the chain length of the organic spacer alters the density, period thickness, and the sound speed within this medium, all of which influence the phonon dispersion as modeled by Eq. (1). The changes in density are relatively minute and do not appreciably alter the calculated dispersion within the mini-Brillouin zone. The increases in both sound velocity and period thickness, on the other hand, compete in influencing the characteristic frequency (v2/d2) of the organic subphase and consequently affect the expected frequency of the LA phonons. As a result, although increasing the organic chain length from four to five carbons increases the period thickness, it also leads to a much larger increase in the sound velocity through the organic medium—enough to considerably decrease the characteristic frequency within the organic subphase—decreasing the number of branches within the mini-Brillouin zone (Fig. S9). Additionally, further increasing the chain length from 5 to 6 carbons, which leads to an increase in the chain length but does not lead to a significant increase in the sound velocity, actually results in a decrease in the expected frequency of the mode. For both of these materials, the Rytov model predicts the existence of a zone-folded LA mode in excellent agreement with our experimental observations (Table II). Importantly, these observations highlight the manner in which the vibrational and structural properties of the organic subphase can uniquely control the vibrational properties of 2D layered materials without simultaneously influencing the optical properties.

TABLE II.

Summary of most prominent zone-folded longitudinal acoustic (zf-LA) and optical Raman mode frequencies for each of the materials studied. All mode frequencies are in units of cm−1.

N = 2 BA:MAPbIn = 2 BA:FAPbIn = 3 BA:MAPbIN = 4 BA:MAPbIn = 2 PA:MAPbIn = 2 HA:MAPbI
zf-LA 25.2 ± 0.1 24.8 ± 0.2 26.7 ± 0.2 26.8 ± 0.3 26.1 ± 0.1 24.3 ± 0.3 
Optical 47.5 ± 0.1 48.5 ± 0.3 47.4 ± 0.2 46.4 ± 0.2 48.6 ± 0.1 46.2 ± 0.2 
N = 2 BA:MAPbIn = 2 BA:FAPbIn = 3 BA:MAPbIN = 4 BA:MAPbIn = 2 PA:MAPbIn = 2 HA:MAPbI
zf-LA 25.2 ± 0.1 24.8 ± 0.2 26.7 ± 0.2 26.8 ± 0.3 26.1 ± 0.1 24.3 ± 0.3 
Optical 47.5 ± 0.1 48.5 ± 0.3 47.4 ± 0.2 46.4 ± 0.2 48.6 ± 0.1 46.2 ± 0.2 

Figure 6(a) and Figs. S4–S8 show the temperature-dependent Raman spectra of all investigated samples. As the temperature increases across the melting transition of the organic subphase, the phonon energies soften discontinuously and the linewidth increases dramatically; this occurs in concert with the effects of a larger central peak in the high temperature phase. For example, such discontinuous changes are visible across the organic subphase melting transition in n = 2 BA:MAPbI at 283 K [Fig. 6(a) between traces at 273 K and 300 K]. Across solid–solid phase transitions, such as that observed for n = 2 PA:MAPbI between 200 K and 250 K, pronounced mode softening occurs gradually over the temperature range associated with the transition (Fig. S4). As observed for the synthetically controlled structural changes discussed above, in each of these cases, the deviation in phonon energy is likely the result of changes in the symmetry and bond lengths of the atoms associated with the said transitions.

FIG. 6.

(a) Low-frequency Raman spectrum of n = 2 BA:MAPbI at various temperatures between room temperature and 77 K. (b) Phonon mode softening for the primary optical mode. (c) Phonon mode softening for the zf-LA mode. (d) Temperature dependence of FWHM for the primary optical mode. (e) Temperature dependence of FWHM for the zf-LA mode.

FIG. 6.

(a) Low-frequency Raman spectrum of n = 2 BA:MAPbI at various temperatures between room temperature and 77 K. (b) Phonon mode softening for the primary optical mode. (c) Phonon mode softening for the zf-LA mode. (d) Temperature dependence of FWHM for the primary optical mode. (e) Temperature dependence of FWHM for the zf-LA mode.

Close modal

In addition to the pronounced spectral changes observed across phase transitions within the various 2D LHPs, we also observe temperature-dependent phonon energy and linewidth shifts in the absence of such transformations [Figs. 6(b)–6(e)]. Each phonon mode discussed above exhibits the same trend, phonon mode softening (red shift) as the temperature of the lattice is increased, along with a slight inflation of the measured peak linewidth. Mode softening with increasing temperature is a widely reported phenomenon, typically attributed to anharmonic lattice characteristics such as thermal expansion. Thermal expansion is observed via powder x-ray diffraction for 2D LHPs,52 and the lengthening of each bond in the lattice naturally leads to a lower force constant for each phonon mode in the Raman spectrum, hence a lower measured frequency. Within the Rytov model for the zone-folded LA phonons, this same behavior is captured via changes in the density and elastic parameters of the lattice.

The increase in the linewidth as a function of temperature is somewhat unexpected, given the energy of the phonon modes studied herein are all well below kBT, wherein the decay of phonons into lower energy phonons ceases to become temperature dependent. Lead halide perovskites, however, exhibit a significant degree of homogeneous broadening owing to the multitude of degrees of freedom available to the organic cations employed, the extent of which is temperature dependent.36 At higher temperatures, increasing degrees of freedom for cation rotation are unlocked, leading to a wider continuum of energies distributed throughout the lattice and, thus, a broader observed Raman mode. It should be reiterated, however, that our observed Raman modes are all best fit by Gaussian line shapes, implying that inhomogeneous broadening due to the variable orientation of the different organic cations is largely responsible for the measured peak width. In the absence of a phase transition such as those described above, this broadening should not be temperature dependent.

Despite the consistency of the Rytov model in predicting the experimentally observed Raman modes below 30 cm−1, we emphasize that the assignment is not conclusive. For crystals with such large unit cell volumes (N = 102 atoms/cell for n = 2 BA:MAPbI), there are a correspondingly large number of optical phonon branches (3N − 3 = 303 optical modes in n = 2 BA:MAPbI). As shown in Fig. S1, DFT necessarily provides many candidate modes, making the conventional assignment quite difficult. The Rytov model provides a compelling alternative picture for viewing the lowest frequency vibrations observed, where DFT does not provide any strong candidates for assignment, but further work is required to fully validate this finding.

It is worth noting that although this behavior is analogous to the behavior previously observed in other periodic layered superlattice-type materials such as all-inorganic superlattices, it is different from the collective vibrations of isolated nanostructures. 2D CdSe nanoplatelets, for instance, exhibit breathing modes of the inorganic platelet that are effectively dampened by the organic surface-bound ligands, much like what is observed for other colloidal CdSe nanoparticles of varying geometry.82,83 In these materials, the inorganic crystal exhibits an acoustic mode the frequency of which is modulated by the organic ligands. 2D LHPs of the type studied herein, on the other hand, appear to exhibit collective vibrations of the entire periodic structure. This is to say, without the presence of any organic subphase (or any interstitial elastic medium), the zf-LA modes assigned herein would presumably cease to be observed. Additionally, whereas breathing modes of the type observed in semiconductor nanoplatelets are extremely and increasingly dependent on the monolayer thickness of the platelet as it decreases below 5 (noted herein using the variable, n), the modes observed for 2D LHPs are more modestly affected by changes in n. The distinctions provide interesting avenues for future work on perovskite nanoplatelets to further study the potential existence of breathing modes in 2D LHPs and to corroborate/extend the limits of the behavior assigned herein.

On the one hand, our results indicate the similarity between 2D LHPs and their bulk counterparts, as all optical modes observed are analogs of bulk phonons, with minimal changes resulting from the additional organic cation and nanostructured length scale. This supports the hypothesis that 2D LHPs are truly hybrid materials in which the organic spacer subphase, much like the A-site cation, does not fundamentally alter the dispersion relation of the PbI inorganic cages. Instead, the vibrational density of states for 2D LHPs more closely resembles the sum of each of these phases, as has been observed for other organic–inorganic hybrid nanomaterials.84 

Our observation of ostensibly coherent acoustic phonons of the layered superlattice, on the other hand, suggests that although 2D LHPs are analogous to their bulk counterparts in many ways, they may be capable of supporting acoustic phonons not accessible to ternary 3D bulk perovskites, such as MAPbI3. These phonons may prove important for understanding a variety of phenomena related to heat transport in these nanomaterials. For instance, as has been observed for inorganic superlattices that exhibit zone-folded LA phonons, thermal transport in these systems may proceed by wave-like phonon transport through which these long wavelength phonons scatter off of one another without regard to the organic–inorganic interfaces that usually limit hybrid organic–inorganic nanostructured semiconductors.85 

See the supplementary material for DFT band dispersions; peak fitting; photographs of 2D perovskite crystals; temperature-dependent Raman spectra for all materials studied; and Rytov model calculations for select compounds.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors would like to thank Katie Mauck for helpful discussions. This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0019345. DFT calculations were supported by Toyota Research Institute (TRI), and computational support was provided by the Extreme Science and Engineering Discovery Environment, supported by the National Science Foundation (Grant No. ACI-1053575). N.D. was supported by the MIT Energy Initiative Society of Energy Fellows.

The authors declare no competing financial interest.

1.
M.
Liu
,
M. B.
Johnston
, and
H. J.
Snaith
,
Nature
501
,
395
(
2013
).
2.
Q.
Chen
,
H.
Zhou
,
Z.
Hong
,
S.
Luo
,
H.-S.
Duan
,
H.-H.
Wang
,
Y.
Liu
,
G.
Li
, and
Y.
Yang
,
J. Am. Chem. Soc.
136
,
622
(
2014
).
3.
H.
Tan
,
A.
Jain
,
O.
Voznyy
,
X.
Lan
,
F. P.
García de Arquer
,
J. Z.
Fan
,
R.
Quintero-Bermudez
,
M.
Yuan
,
B.
Zhang
,
Y.
Zhao
,
F.
Fan
,
P.
Li
,
L. N.
Quan
,
Y.
Zhao
,
Z.-H.
Lu
,
Z.
Yang
,
S.
Hoogland
, and
E. H.
Sargent
,
Science
355
,
722
(
2017
).
4.
J.
Burschka
,
N.
Pellet
,
S.-J.
Moon
,
R.
Humphry-Baker
,
P.
Gao
,
M. K.
Nazeeruddin
, and
M.
Grätzel
,
Nature
499
,
316
(
2013
).
5.
H.
Zhou
,
Q.
Chen
,
G.
Li
,
S.
Luo
,
T.-b.
Song
,
H.-S.
Duan
,
Z.
Hong
,
J.
You
,
Y.
Liu
, and
Y.
Yang
,
Science
345
,
542
(
2014
).
6.
Y.-H.
Kim
,
H.
Cho
,
J. H.
Heo
,
T.-S.
Kim
,
N.
Myoung
,
C.-L.
Lee
,
S. H.
Im
, and
T.-W.
Lee
,
Adv. Mater.
27
,
1248
(
2015
).
7.
M.
Yuan
,
L. N.
Quan
,
R.
Comin
,
G.
Walters
,
R.
Sabatini
,
O.
Voznyy
,
S.
Hoogland
,
Y.
Zhao
,
E. M.
Beauregard
,
P.
Kanjanaboos
,
Z.
Lu
,
D. H.
Kim
, and
E. H.
Sargent
,
Nat. Nanotechnol.
11
,
872
(
2016
).
8.
M.
Zhang
,
F.
Zhang
,
Y.
Wang
,
L.
Zhu
,
Y.
Hu
,
Z.
Lou
,
Y.
Hou
, and
F.
Teng
,
Sci. Rep.
8
,
11157
(
2018
).
9.
A.
Mei
,
X.
Li
,
L.
Liu
,
Z.
Ku
,
T.
Liu
,
Y.
Rong
,
M.
Xu
,
M.
Hu
,
J.
Chen
,
Y.
Yang
,
M.
Grätzel
, and
H.
Han
,
Science
345
,
295
(
2014
).
10.
A.
Miyata
,
A.
Mitioglu
,
P.
Plochocka
,
O.
Portugall
,
J. T.-W.
Wang
,
S. D.
Stranks
,
H. J.
Snaith
, and
R. J.
Nicholas
,
Nat. Phys.
11
,
582
(
2015
).
11.
W.
Nie
,
J. C.
Blancon
,
A. J.
Neukirch
,
K.
Appavoo
,
H.
Tsai
,
M.
Chhowalla
,
M. A.
Alam
,
M. Y.
Sfeir
,
C.
Katan
,
J.
Even
,
S.
Tretiak
,
J. J.
Crochet
,
G.
Gupta
, and
A. D.
Mohite
,
Nat. Commun.
7
,
11574
(
2016
).
12.
W.
Huang
,
J. S.
Manser
,
P. V.
Kamat
, and
S.
Ptasinska
,
Chem. Mater.
28
,
303
(
2016
).
13.
D. B.
Mitzi
,
S.
Wang
,
C. A.
Feild
,
C. A.
Chess
, and
A. M.
Guloy
,
Science
267
,
1473
(
1995
).
14.
D. B.
Mitzi
,
Chem. Mater.
8
,
791
(
1996
).
15.
D. B.
Mitzi
,
C. D.
Dimitrakopoulos
, and
L. L.
Kosbar
,
Chem. Mater.
13
,
3728
(
2001
).
16.
I. C.
Smith
,
E. T.
Hoke
,
D.
Solis-Ibarra
,
M. D.
McGehee
, and
H. I.
Karunadasa
,
Angew. Chem. Int. Ed.
53
,
11232
(
2014
).
17.
J.-C.
Blancon
,
H.
Tsai
,
W.
Nie
,
C. C.
Stoumpos
,
L.
Pedesseau
,
C.
Katan
,
M.
Kepenekian
,
C. M. M.
Soe
,
K.
Appavoo
,
M. Y.
Sfeir
,
S.
Tretiak
,
P. M.
Ajayan
,
M. G.
Kanatzidis
,
J.
Even
,
J. J.
Crochet
, and
A. D.
Mohite
,
Science
355
,
1288
(
2017
).
18.
M. D.
Smith
,
L.
Pedesseau
,
M.
Kepenekian
,
I. C.
Smith
,
C.
Katan
,
J.
Even
, and
H. I.
Karunadasa
,
Chem. Sci.
8
,
1960
(
2017
).
19.
M. D.
Smith
,
E. J.
Crace
,
A.
Jaffe
, and
H. I.
Karunadasa
,
Annu. Rev. Mater. Res.
48
,
111
(
2018
).
20.
B.
Dhanabalan
,
A.
Castelli
,
M.
Palei
,
D.
Spirito
,
L.
Manna
,
R.
Krahne
, and
M.
Arciniegas
,
Nanoscale
11
,
8334
(
2019
).
21.
C. M.
Mauck
and
W. A.
Tisdale
,
Trends Chem.
1
,
380
(
2019
).
22.
A. R.
Srimath Kandada
and
C.
Silva
,
J. Phys. Chem. Lett.
11
,
3173
(
2020
).
23.
J.-C.
Blancon
,
A. V.
Stier
,
H.
Tsai
,
W.
Nie
,
C. C.
Stoumpos
,
B.
Traoré
,
L.
Pedesseau
,
M.
Kepenekian
,
F.
Katsutani
,
G. T.
Noe
,
J.
Kono
,
S.
Tretiak
,
S. A.
Crooker
,
C.
Katan
,
M. G.
Kanatzidis
,
J. J.
Crochet
,
J.
Even
, and
A. D.
Mohite
,
Nat. Commun.
9
,
2254
(
2018
).
24.
D. H.
Cao
,
C. C.
Stoumpos
,
O. K.
Farha
,
J. T.
Hupp
, and
M. G.
Kanatzidis
,
J. Am. Chem. Soc.
137
,
7843
(
2015
).
25.
H.
Tsai
,
W.
Nie
,
J.-C.
Blancon
,
C. C.
Stoumpos
,
R.
Asadpour
,
B.
Harutyunyan
,
A. J.
Neukirch
,
R.
Verduzco
,
J. J.
Crochet
,
S.
Tretiak
,
L.
Pedesseau
,
J.
Even
,
M. A.
Alam
,
G.
Gupta
,
J.
Lou
,
P. M.
Ajayan
,
M. J.
Bedzyk
,
M. G.
Kanatzidis
, and
A. D.
Mohite
,
Nature
536
,
312
(
2016
).
26.
Z.
Wang
,
Q.
Lin
,
F. P.
Chmiel
,
N.
Sakai
,
L. M.
Herz
, and
H. J.
Snaith
,
Nat. Energy
2
,
17135
(
2017
).
27.
G.
Grancini
,
C.
Roldán-Carmona
,
I.
Zimmermann
,
E.
Mosconi
,
X.
Lee
,
D.
Martineau
,
S.
Narbey
,
F.
Oswald
,
F.
De Angelis
,
M.
Graetzel
, and
M. K.
Nazeeruddin
,
Nat. Commun.
8
,
15684
(
2017
).
28.
P.
Li
,
Y.
Zhang
,
C.
Liang
,
G.
Xing
,
X.
Liu
,
F.
Li
,
X.
Liu
,
X.
Hu
,
G.
Shao
, and
Y.
Song
,
Adv. Mater.
30
,
1805323
(
2018
).
29.
X.
Yang
,
X.
Zhang
,
J.
Deng
,
Z.
Chu
,
Q.
Jiang
,
J.
Meng
,
P.
Wang
,
L.
Zhang
,
Z.
Yin
, and
J.
You
,
Nat. Commun.
9
,
570
(
2018
).
30.
D.
Liang
,
Y.
Peng
,
Y.
Fu
,
M. J.
Shearer
,
J.
Zhang
,
J.
Zhai
,
Y.
Zhang
,
R. J.
Hamers
,
T. L.
Andrew
, and
S.
Jin
,
ACS Nano
10
,
6897
(
2016
).
31.
Z.
Tan
,
Y.
Wu
,
H.
Hong
,
J.
Yin
,
J.
Zhang
,
L.
Lin
,
M.
Wang
,
X.
Sun
,
L.
Sun
,
Y.
Huang
,
K.
Liu
,
Z.
Liu
, and
H.
Peng
,
J. Am. Chem. Soc.
138
,
16612
(
2016
).
32.
A.
Castelli
,
G.
Biffi
,
L.
Ceseracciu
,
D.
Spirito
,
M.
Prato
,
D.
Altamura
,
C.
Giannini
,
S.
Artyukhin
,
R.
Krahne
,
L.
Manna
, and
M. P.
Arciniegas
,
Adv. Mater.
31
,
1805608
(
2019
).
33.
M.
Ledinský
,
P.
Löper
,
B.
Niesen
,
J.
Holovský
,
S.-J.
Moon
,
J.-H.
Yum
,
S.
De Wolf
,
A.
Fejfar
, and
C.
Ballif
,
J. Phys. Chem. Lett.
6
,
401
(
2015
).
34.
C. M.
Iaru
,
J. J.
Geuchies
,
P. M.
Koenraad
,
D.
Vanmaekelbergh
, and
A. Y.
Silov
,
ACS Nano
11
,
11024
(
2017
).
35.
D. B.
Straus
,
S.
Hurtado Parra
,
N.
Iotov
,
J.
Gebhardt
,
A. M.
Rappe
,
J. E.
Subotnik
,
J. M.
Kikkawa
, and
C. R.
Kagan
,
J. Am. Chem. Soc.
138
,
13798
(
2016
).
36.
A. M. A.
Leguy
,
A. R.
Goñi
,
J. M.
Frost
,
J.
Skelton
,
F.
Brivio
,
X.
Rodríguez-Martínez
,
O. J.
Weber
,
A.
Pallipurath
,
M. I.
Alonso
,
M.
Campoy-Quiles
,
M. T.
Weller
,
J.
Nelson
,
A.
Walsh
, and
P. R. F.
Barnes
,
Phys. Chem. Chem. Phys.
18
,
27051
(
2016
).
37.
C.
Quarti
,
G.
Grancini
,
E.
Mosconi
,
P.
Bruno
,
J. M.
Ball
,
M. M.
Lee
,
H. J.
Snaith
,
A.
Petrozza
, and
F.
De Angelis
,
J. Phys. Chem. Lett.
5
,
279
(
2014
).
38.
O.
Yaffe
,
Y.
Guo
,
L. Z.
Tan
,
D. A.
Egger
,
T.
Hull
,
C. C.
Stoumpos
,
F.
Zheng
,
T. F.
Heinz
,
L.
Kronik
,
M. G.
Kanatzidis
,
J. S.
Owen
,
A. M.
Rappe
,
M. A.
Pimenta
, and
L. E.
Brus
,
Phys. Rev. Lett.
118
,
136001
(
2017
).
39.
F.
Brivio
,
J. M.
Frost
,
J. M.
Skelton
,
A. J.
Jackson
,
O. J.
Weber
,
M. T.
Weller
,
A. R.
Goñi
,
A. M. A.
Leguy
,
P. R. F.
Barnes
, and
A.
Walsh
,
Phys. Rev. B
92
,
144308
(
2015
).
40.
T.
Ivanovska
,
C.
Quarti
,
G.
Grancini
,
A.
Petrozza
,
F.
De Angelis
,
A.
Milani
, and
G.
Ruani
,
ChemSusChem
9
,
2994
(
2016
).
41.
G.
Batignani
,
G.
Fumero
,
A. R.
Srimath Kandada
,
G.
Cerullo
,
M.
Gandini
,
C.
Ferrante
,
A.
Petrozza
, and
T.
Scopigno
,
Nat. Commun.
9
,
1971
(
2018
).
42.
M.
Park
,
A. J.
Neukirch
,
S. E.
Reyes-Lillo
,
M.
Lai
,
S. R.
Ellis
,
D.
Dietze
,
J. B.
Neaton
,
P.
Yang
,
S.
Tretiak
, and
R. A.
Mathies
,
Nat. Commun.
9
,
2525
(
2018
).
43.
M. A.
Pérez-Osorio
,
R. L.
Milot
,
M. R.
Filip
,
J. B.
Patel
,
L. M.
Herz
,
M. B.
Johnston
, and
F.
Giustino
,
J. Phys. Chem. C
119
,
25703
(
2015
).
44.
M. A.
Pérez-Osorio
,
Q.
Lin
,
R. T.
Phillips
,
R. L.
Milot
,
L. M.
Herz
,
M. B.
Johnston
, and
F.
Giustino
,
J. Phys. Chem. C
122
,
21703
(
2018
).
45.
Y.
Abid
,
J. Phys.: Condens. Matter
6
,
6447
(
1994
).
46.
M.
Couzi
,
A.
Daoud
, and
R.
Perret
,
Phys. Status Solidi A
41
,
271
(
1977
).
47.
N.
Preda
,
L.
Mihut
,
M.
Baibarac
,
I.
Baltog
, and
S.
Lefrant
,
J. Phys.: Condens. Matter
18
,
8899
(
2006
).
48.
R. F.
Warren
and
W. Y.
Liang
,
J. Phys.: Condens. Matter
5
,
6407
(
1993
).
49.
C. M.
Mauck
,
A.
France-Lanord
,
A. C.
Hernandez Oendra
,
N. S.
Dahod
,
J. C.
Grossman
, and
W. A.
Tisdale
,
J. Phys. Chem. C
123
,
27904
(
2019
).
50.
B.
Dhanabalan
,
Y.-C.
Leng
,
G.
Biffi
,
M.-L.
Lin
,
P.-H.
Tan
,
I.
Infante
,
L.
Manna
,
M. P.
Arciniegas
, and
R.
Krahne
,
ACS Nano
14
,
4689
(
2020
).
51.
F.
Thouin
,
D. A.
Valverde-Chávez
,
C.
Quarti
,
D.
Cortecchia
,
I.
Bargigia
,
D.
Beljonne
,
A.
Petrozza
,
C.
Silva
, and
A. R.
Srimath Kandada
,
Nat. Mater.
18
,
349
(
2019
).
52.
W.
Paritmongkol
,
N. S.
Dahod
,
A.
Stollmann
,
N.
Mao
,
C.
Settens
,
S.-L.
Zheng
, and
W. A.
Tisdale
,
Chem. Mater.
31
,
5592
(
2019
).
53.
P.
Hohenberg
and
W.
Kohn
,
Phys. Rev.
136
,
B864
(
1964
).
54.
W.
Kohn
and
L. J.
Sham
,
Phys. Rev.
137
,
A1697
(
1965
).
55.
G.
Kresse
and
J.
Hafner
,
Phys. Rev. B
48
,
13115
(
1993
).
56.
G.
Kresse
and
J.
Furthmüller
,
Comput. Mater. Sci.
6
,
15
(
1996
).
57.
MedeA-2.22, Materials Design, Inc., San Diego, CA, USA (
2018
).
58.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
59.
P. E.
Blöchl
,
Phys. Rev. B
50
,
17953
(
1994
).
60.
S.
Grimme
,
J.
Antony
,
S.
Ehrlich
, and
H.
Krieg
,
J. Chem. Phys.
132
,
154104
(
2010
).
61.
K.
Parlinski
,
Z. Q.
Li
, and
Y.
Kawazoe
,
Phys. Rev. Lett.
81
,
3298
(
1997
).
62.
D.
Porezag
and
M. R.
Pederson
,
Phys. Rev. B
54
,
7830
(
1996
).
63.
M.
Gajdoš
,
K.
Hummer
,
G.
Kresse
,
J.
Furthmüller
, and
F.
Bechstedt
,
Phys. Rev. B
73
,
045112
(
2006
).
64.
A. A.
Bakulin
,
O.
Selig
,
H. J.
Bakker
,
Y. L. A.
Rezus
,
C.
Müller
,
T.
Glaser
,
R.
Lovrincic
,
Z.
Sun
,
Z.
Chen
,
A.
Walsh
,
J. M.
Frost
, and
T. L. C.
Jansen
,
J. Phys. Chem. Lett.
6
,
3663
(
2015
).
65.
C. C.
Stoumpos
,
D. H.
Cao
,
D. J.
Clark
,
J.
Young
,
J. M.
Rondinelli
,
J. I.
Jang
,
J. T.
Hupp
, and
M. G.
Kanatzidis
,
Chem. Mater.
28
,
2852
(
2016
).
66.
N. S.
Dahod
,
W.
Paritmongkol
,
A.
Stollmann
,
C.
Settens
,
S.-L.
Zheng
, and
W. A.
Tisdale
,
J. Phys. Chem. Lett.
10
,
2924
(
2019
).
67.
A. S.
Barker
,
J. L.
Merz
, and
A. C.
Gossard
,
Phys. Rev. B
17
,
3181
(
1978
).
68.
C.
Colvard
,
R.
Merlin
,
M. V.
Klein
, and
A. C.
Gossard
,
Phys. Rev. Lett.
45
,
298
(
1980
).
69.
C.
Colvard
,
T. A.
Gant
,
M. V.
Klein
,
R.
Merlin
,
R.
Fischer
,
H.
Morkoc
, and
A. C.
Gossard
,
Phys. Rev. B
31
,
2080
(
1985
).
70.
A.
Bartels
,
T.
Dekorsy
,
H.
Kurz
, and
K.
Köhler
,
Phys. Rev. Lett.
82
,
1044
(
1999
).
71.
J.
Ibáñez
,
A.
Rapaport
,
C.
Boney
,
R.
Oliva
,
R.
Cuscó
,
A.
Bensaoula
, and
L.
Artús
,
J. Raman Spectrosc.
43
,
237
(
2012
).
72.
D.
Schneider
,
F.
Liaqat
,
E. H.
El Boudouti
,
Y.
El Hassouani
,
B.
Djafari-Rouhani
,
W.
Tremel
,
H.-J.
Butt
, and
G.
Fytas
,
Nano Lett.
12
,
3101
(
2012
).
73.
S. M.
Rytov
,
Sov. Phys. - Acoust.
2
,
68
(
1956
).
74.
P.
Guo
,
C. C.
Stoumpos
,
L.
Mao
,
S.
Sadasivam
,
J. B.
Ketterson
,
P.
Darancet
,
M. G.
Kanatzidis
, and
R. D.
Schaller
,
Nat. Commun.
9
,
2019
(
2018
).
75.
A.
Gold-Parker
,
P. M.
Gehring
,
J. M.
Skelton
,
I. C.
Smith
,
D.
Parshall
,
J. M.
Frost
,
H. I.
Karunadasa
,
A.
Walsh
, and
M. F.
Toney
,
Proc. Natl. Acad. Sci. U. S. A.
115
,
11905
(
2018
).
76.
S.-Y.
Yue
,
X.
Zhang
,
G.
Qin
,
J.
Yang
, and
M.
Hu
,
Phys. Rev. B
94
,
115427
(
2016
).
77.
W.
Lee
,
H.
Li
,
A. B.
Wong
,
D.
Zhang
,
M.
Lai
,
Y.
Yu
,
Q.
Kong
,
E.
Lin
,
J. J.
Urban
,
J. C.
Grossman
, and
P.
Yang
,
Proc. Natl. Acad. Sci. U. S. A.
114
,
8693
(
2017
).
78.
A. C.
Ferreira
,
A.
Létoublon
,
S.
Paofai
,
S.
Raymond
,
C.
Ecolivet
,
B.
Rufflé
,
S.
Cordier
,
C.
Katan
,
M. I.
Saidaminov
,
A. A.
Zhumekenov
,
O. M.
Bakr
,
J.
Even
, and
P.
Bourges
,
Phys. Rev. Lett.
121
,
085502
(
2018
).
79.
M.
Fu
,
P.
Tamarat
,
J.-B.
Trebbia
,
M. I.
Bodnarchuk
,
M. V.
Kovalenko
,
J.
Even
, and
B.
Lounis
,
Nat. Commun.
9
,
3318
(
2018
).
80.
G. A.
Elbaz
,
W.-L.
Ong
,
E. A.
Doud
,
P.
Kim
,
D. W.
Paley
,
X.
Roy
, and
J. A.
Malen
,
Nano Lett.
17
,
5734
(
2017
).
81.
C. C.
Stoumpos
,
C. D.
Malliakas
, and
M. G.
Kanatzidis
,
Inorg. Chem.
52
,
9019
(
2013
).
82.
E. M. Y.
Lee
,
A. J.
Mork
,
A. P.
Willard
, and
W. A.
Tisdale
,
J. Chem. Phys.
147
,
044711
(
2017
).
83.
A.
Girard
,
L.
Saviot
,
S.
Pedetti
,
M. D.
Tessier
,
J.
Margueritat
,
H.
Gehan
,
B.
Mahler
,
B.
Dubertret
, and
A.
Mermet
,
Nanoscale
8
,
13251
(
2016
).
84.
W.-L.
Ong
,
S. M.
Rupich
,
D. V.
Talapin
,
A. J. H.
McGaughey
, and
J. A.
Malen
,
Nat. Mater.
12
,
410
(
2013
).
85.
B.
Yang
and
G.
Chen
,
Phys. Rev. B
67
,
195311
(
2003
).

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