Metal cation s lone-pairs increase octahedral tilting instabilities in halide perovskites

Abstract

Halide perovskites exhibit beneficial opto-electronic properties (e.g. long carrier lifetimes and low defect densities), and their dynamic structural instabilities and anharmonic thermal fluctuations are directly implicated in these properties. In this work, we combine in-depth analysis of Raman spectroscopy and ab initio calculations to uncover the critical roles of Group 14 M²⁺ (M = Pb, Sn, Ge) metal cation s orbital lone pairs in the dynamic instabilities of CsMBr₃ and particularly in governing the octahedral tilting. Previous studies concluded that the lone-pair stereochemical activity primarily leads to the off-centering motion of the metal cation, and the tilting is usually ascribed to ionic size effects. Here, we show that the lone-pair stereochemical activity contributes to strong octahedral tilting instabilities that induce liquid-like behavior in all examined crystals, which underlies the robustness of halide perovskites to charged defects. In addition, the lone-pairs induce a local, molecule-like behavior of the Ge²⁺ with a pyramidal bonding motif in the cubic phase, and they contribute to another phase transition of CsSnBr₃ at 60 K. Our findings elucidate the fundamental origin of anharmonicities in halide perovskites and provide the crucial link between chemical composition and optoelectronic properties, opening opportunities for lead-free and solution-processable photovoltaics.

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Introduction

Halide perovskites (HPs) are extensively studied as potential replacements for the traditional photovoltaic materials.¹ Lead-based HPs have demonstrated solar power conversion efficiencies surpassing 25% in thin films and 29% in tandem cells with Si.² This motivates recent efforts to develop new photovoltaic cells that are based on nontoxic perovskite halides, where Pb²⁺ is replaced with Sn²⁺ and/or Ge²⁺.^3–7

From a fundamental standpoint, the fascinating opto-electronic properties of HPs are closely related to their highly anharmonic thermal motions that lead to local, polar fluctuations and thus, localization of charge carriers.^8–11,75 The perovskite crystal (with stoichiometry AMX₃) consists of a three-dimensional network of corner-sharing MX₆ octahedra, with A-site cations occupying the cuboctahedral voids. The high connectivity allows for various lower-symmetry phases, characterized by tilting of the octahedra, making HPs capable of accommodating a wide range of compositions.^12,13 This compositional flexibility enables a plethora of phenomena, tuned by competing short- and long–range anharmonic couplings, leading to polar, magnetic, and tilting phase transitions.^14–18

Geometric or ionic (hard sphere packing) models, which are based only on the sizes of the constituent ions (r_A,M,X), hold great predictive power for whether a perovskite will form.^19–23 The tolerance factor, of Goldschmidt¹⁹ is the most successful geometric parameter for perovskite prediction: if t > 1, then the A-site cation is too large to fit in the MX₆ cuboctahedral cavities, and that disfavors the formation of a perovskite. If t < 1, then the A-site cation is relatively small for the interstitial region between octahedra, and so the corner-sharing octahedra tilt to fill the space. Following different tilting patterns, many low-symmetry structures can be formed due to this tilting instability.^21–31 The tilting can also be dynamic, resulting in structural fluctuations.^{8,9,14,32–36} When t is too low, the corner-sharing octahedra prefer to rearrange into edge-sharing or face-sharing octahedra,³⁷ taking the system away from the perovskite structure. In addition to t, the octahedral factor μ = r_M/r_X is another important geometric factor. According to Pauling's first rule,²⁰μ should be large enough to guarantee that the M cation makes contact with all surrounding X anions. Recent studies construct structural maps vs. t and μ and correctly categorize oxides and halides as perovskites or non-perovskites at room temperature.^38–42

Notably, these geometric models represent an ionic limit and contain no information about covalent bonding. Therefore, they sometimes fail to predict the correct structural phase evolution of compounds with certain electron configurations and bonding preferences.^43,44 Heavier halogen ions have lower electronegativity and stronger covalency, reducing the accuracy and applicability of the hard-sphere ionic models.^38–41 Furthermore, main-group cations in lower oxidation states (e.g. Pb²⁺, Sn²⁺, Ge²⁺) retain their outermost s electrons as a nonbonded lone pair.^45–48 This lone pair occupies space, behaves like an additional ligand, and is said to be stereochemically active. Acentric distortions of the coordination environments induced by such lone-pair cations allow an energetically favorable mixing with the ligand (anion) orbitals which is symmetry-forbidden in the undistorted state.^49–53 In oxides such as PbTiO₃, Pb(Zr_1−xTi_x)O₃, lone-pair cations reside on the A site. The large polar displacements of A cations caused by lone-pair effects give rise to the large polarization and giant piezoelectric responses.^54–58 In HPs, they occupy the M site, and significant M off-centering motions are observed experimentally and computationally.^9,46,59–66 Importantly, previous studies show in HPs off-centering instabilities induced by lone-pairs and tilting instabilities can coexist and are competing with each other.^67,68 The tendency for lone-pair-driven distortion is stronger for lighter cations.⁵⁰ In PbTiO₃-based solid solutions, the overall effects of both instabilities have been used to predict the morphotropic phase boundary (MPB) composition and the transition temperature at the MPB.⁶⁹

Here, we aim to study the interplay between geometric (ionic size) and lone-pair (covalency) effects on the structural dynamics of halide perovskites. To that end, we combine Raman spectroscopy and ab initio molecular dynamics to compare the structural dynamics of CsMBr₃ in their cubic (with M = Pb, Sn, and Ge) and orthorhombic (M = Pb, Sn) phases. This chemical series tunes both the energy scale of the lone-pair-driven distortions and the relative ionic sizes which are the basis for the geometric models. Therefore, changing the central M cation provides an excellent means of establishing the relative importance of concurrent size and covalence effects in the structural dynamics of the HPs. In our analysis, we go beyond the harmonic approximation^68,70 as we assign real-space motion from ab initio molecular dynamics to the experimentally observed spectral features.

Our main new finding is that the lone pair contributes significantly to octahedral tilting instabilities that go beyond what is predicted by geometric models. These fluctuations give rise to strong, diffuse inelastic light scattering that increases towards 0 cm⁻¹ (“central peak”) that is similar to that of liquids. In addition to tilting, the strong lone-pair stereochemical activity also induces dynamic Ge²⁺ off-centering motions as dynamic fluctuations between pyramidal environments in CsGeBr₃ that are uncorrelated to neighbors, leading to a molecule-like behavior. At around 60 K, in CsSnBr₃ the lone pair contributes to another phase transition to lower symmetry phase. We expect these lone-pair driven structural instabilities discovered in CsMBr₃ also prevail in their organic HP counterparts, which are more widely used in photovoltaics.⁷⁶

Results and discussion

We begin our study by comparing the structural dynamics of CsPbBr₃, CsSnBr₃, and CsGeBr₃ in their cubic phases (Pm3̄m, space group #221). A detailed discussion of the phase sequence, ionic size and lone-pair stereochemical activity of the three crystals is given in the section “Inadequacy of ionic size models” of ESI.† It is important to note that based on the symmetry of average structure, the crystals should be Raman inactive.⁷¹ Yet, as shown in Fig. 1(a)–(c) all three crystals show strong, first order Raman scattering. The failure of factor group analysis to predict the Raman spectra is an important indication that the thermal fluctuations of all three crystals are strongly anharmonic. The dash-and-dot lines show the spectral features that are extracted from a fit of each spectrum to the product of the Bose–Einstein distribution and a sum of damped Lorentz oscillators (see ESI† for fitting procedures).

Next, we performed ab initio molecular dynamics (AIMD) at relevant temperatures (500 K, 330 K and 663 K for CsPbBr₃, CsSnBr₃, and CsGeBr₃, respectively) and calculated dynamic Raman spectra based on AIMD trajectories (calculation details are in the Section “computational methods” of ESI†). As shown in Fig. 1, theoretical spectra well reproduce experimental Raman spectra of all the compounds.

The spectra of all three crystals consist of a broad feature increasing continuously toward 0 cm⁻¹ superimposed by broad shoulders from 50–100 cm⁻¹. The low-frequency broad feature is composed of over-damped Lorentz oscillators (width ≈ 50 cm⁻¹). In that sense, their structural dynamics resemble that of a liquid. To demonstrate this point, the spectrum of benzene at room temperature is overlaid in Fig. 1(a) and (b). Contrary to CsPbBr₃ and CsSnBr₃, the Raman spectrum of CsGeBr₃ (Fig. 1(c)) has a dominant spectral feature centered at ≈ 148 cm⁻¹.

To elucidate the origin of Raman activity for the three perovskites, we frequency-filter the AIMD trajectories and project them onto normal-mode coordinates that are presented in Fig. 1(d)–(f). The frequency-filtering method and projection method are introduced in the Section “computational methods” of ESI.† Below 30 cm⁻¹, for all three crystals, apart from the harmonic Cs motion, Br₆ octahedral tiltings are the most dominant motions, expected for CsSnBr₃ and CsPbBr₃ due to their low tolerance factor (t = 0.92 and 0.86 respectively) but are surprising for CsGeBr₃ that has a near ideal t (see ESI† for more details). Therefore, the tilting instabilities cannot be fully reconciled with a geometric model solely from ionic effects. Nevertheless, consistent with its near-ideal tolerance factor, the projection weight of the octahedral tilting in CsGeBr₃ is lower than its counterpart in CsSnBr₃ and CsPbBr₃. It has been shown in CsPbF₃ that the Pb lone pair can determine the tilting pattern and ground-state structure by displacing surrounding anions and modifying the A-site environments.⁶⁷ The clear presence of tilting indicates that apart from ionic effects, the lone pair is another source of such instabilities.

Recent density functional theory (DFT) studies predict that in the cubic phase, tilting instabilities can only emerge in CsSnBr₃ and CsPbBr₃ but not in CsGeBr₃.^68,70 This agrees with the geometric analysis but differs from the present results. In DFT, computational analysis of phonons is generally conducted by finding the harmonic vibrations with respect to a reference structure (e.g. the highly symmetric cubic aristotype). AIMD samples the realistic, multi-dimensional potential energy surface and visits various thermally-accessible configurations dynamically. On a time scale of several hundred femtoseconds, the dynamic tilting describes the reorientation of the acentrically-distorted octahedra, and induces various and distinct polar fluctuations in different parts of the solid-which leads to the solid exhibiting spectroscopic signatures like those of a liquid (Fig. 1(a)–(c)).^9,10

In the middle frequency range (50–100 cm⁻¹), motions are dominated by Br₆ distortion, and its peak is at 50 cm⁻¹. The M motion of CsPbBr₃ is also in this middle frequency range, while in CsSnBr₃ and CsGeBr₃ M motions are at higher frequencies, following a trend inverse to the square root of their masses. As shown in Fig. 1(a)–(c), at 70 cm⁻¹ there are clear shoulders in CsPbBr₃ and CsGeBr₃, and a broader shoulder in CsSnBr₃. By relating the fitted signatures in Fig. 1(a)–(c) with the peak of motions in Fig. 1(d)–(f), we find that these shoulders result from Br₆ distortion-driven vibrational modes for all three crystals.

We now turn to discussing the origin of the broad and strong spectral feature that is observed only in CsGeBr₃ at ≈ 148 cm⁻¹. At high frequencies (>100 cm⁻¹), the M–Br stretch is similar for all three crystals and is significant in a wide frequency range (magenta lines in Fig. 1(d)–(f)). However, the Raman scattering is very weak in CsPbBr₃ and CsSnBr₃. Therefore, we ascribe this feature to the Ge off-centering motion. We can expect that Ge will display much stronger off-centering than Pb and Sn for the following reasons related to mass, size, and bonding. First, Br is much closer in mass to Ge²⁺ than to Sn²⁺ and Pb²⁺, so concerted motions will have more weight on the Ge²⁺ cation. Second, Ge²⁺ is nominally too small for octahedral coordination with Br, so it can rattle within the octahedral void formed by the anions. Lastly, the Ge²⁺ lone pair is strongly stereochemically active, favoring a 3+3 coordination (3 short bonds, 3 long bonds; essentially a [GeBr₃]⁻ pyramid, Fig. S1, ESI†). This relationship with the pyramidal motif is further reinforced by the shape of the ≈ 148 cm⁻¹ spectral feature in CsGeBr₃, which is composed of at least two Lorentz oscillators (Fig. 1(c)). This shape is indicative of the asymmetric stretch ν₃ for a pyramidal molecule with C_3v symmetry which is split by the inversion doubling effect.^72,73 As suggested by Thiele et al.,⁷⁴ the temperature-driven rhombohedral to cubic phase transition is an order–disorder transition, where the cubic long-range ordering is given by the average of these positions but the local structure remains largely pyramidal due to the outsized stereochemical activity of the Ge²⁺ lone pair. The noteworthy high-frequency Raman activity in CsGeBr₃ suggests dynamic fluctuations between pyramidal environments that are locally similar to the rhombohedral ground state, with significant Ge²⁺ displacements.

Fig. 1 Upper Panel: high temperature, cubic phase Raman spectra of CsPbBr₃, CsSnBr₃, and CsGeBr₃ single crystals. Lower Panel: projections of frequency-filtered MD for the three crystals. The dominance of the central peak decreases as the ionic size decreases, indicating that the relaxational tilting has an opposite trend compared with stereochemical activities.

Indeed, from projections of frequency-filtered AIMD, the Ge²⁺ off-centering motions are markedly different from those of Pb²⁺ and Sn²⁺ (yellow trace in Fig. 1(d)–(f)). The latter two show a single-peaked feature within a narrow frequency range, revealing that the M atoms move in a harmonic and collective pattern. As a contrast, CsGeBr₃ exhibits a combination of multiple peaks at different frequencies between ≈ 140–200 cm⁻¹, indicating the Ge motion is more disordered and local on individual atom and thus the symmetry of the whole crystal has been lowered. To further demonstrate the weakly correlated Ge motion, we compute the pair-correlation function of neighbor M atoms from the AIMD trajectory as:

where m and n refer to nearest-neighbor M atoms, and R⃑(t) is the atomic displacement. Fig. 2 shows that at their corresponding frequencies, off-centering motions between neighboring M atoms in CsGeBr₃ are less correlated than in CsSnBr₃ and CsPbBr₃. This indicates that the motion of the Ge²⁺ ion is more local and behaves like a molecule. Combined with above experimental analysis, we ascribe this high-frequency Raman activity to motion of Ge atom in its low-symmetry local environment.

Fig. 2 Computed pair-correlation function of neighbor M atoms in CsPbBr₃, CsSnBr₃, and CsGeBr₃. The arrow denotes M off-centering motions according to their frequencies, identified from Fig. 1. M off-centering motion in CsGeBr₃ is less correlated than in CsPbBr₃ and CsSnBr₃.

Summarizing our findings related to the cubic phase of the crystals, we note two different dynamic fluctuations that are related to lone-pair stereochemical activities. The first is the tilting instability that is present in all crystals and results in liquid-like light scattering at low frequencies. The second is the M-cation off-centering instability that only leads to molecular-like light scattering in CsGeBr₃ at high frequencies. The tilting is stronger in CsSnBr₃ and CsPbBr₃, with less active lone pairs and smaller tolerance factors, while the off-centering is stronger in CsGeBr₃ with more active lone pair and roughly unity tolerance factor. The prominence of these two instabilities is controlled by the interplay of ionic size and covalency effects. Raman signatures of both type of fluctuations are not predicted by models and computations that are based on either the average static structure or the single ground-state structure of these crystals.

We now compare the structural dynamics of the low-temperature orthorhombic phases (Pnma, space group #62) of CsPbBr₃ and CsSnBr₃. This comparison has merit because unlike the cubic phase, this phase has well-defined Raman activity based on its space-group symmetry. CsGeBr₃ is excluded from this comparison because, as mentioned previously, it acquires a rhombohedral structure at low temperature. Fig. 3(a) and (b) present the Raman spectra of CsPbBr₃ and CsSnBr₃ at 80 K, where both crystals are in their orthorhombic phase. The Raman spectrum of CsPbBr₃ has only sharp and well resolved Raman peaks that are fitted well by a damped Lorentz oscillator model. Three groups of peaks are identified at 20 cm⁻¹, 50 cm⁻¹, and 80 cm⁻¹, respectively. In contrast, the spectral deconvolution of the CsSnBr₃ crystal requires a Debye relaxational component (purple solid line in Fig. 3(b)) in addition to the Lorentz oscillators (see ESI†). This relaxational component indicates that the octahedral instabilities are still significant in CsSnBr₃ even at 80 K. Four groups of peaks are identified at 20 cm⁻¹, 40 cm⁻¹, and 60 cm⁻¹, 70 cm⁻¹, respectively. We note that for both compounds, theory and experiments agree very well in terms of the magnitudes and frequencies (±10 cm⁻¹) of these well resolved Raman-active peaks.

Fig. 3 Upper Panel: low temperature, orthorhombic phase Raman spectra of CsPbBr₃, CsSnBr₃ single crystals. Lower Panel: projections of frequency-filtered MD for two crystals. The relaxational tilting motion is only present in CsSnBr₃ with a larger tolerance factor than that of CsPbBr₃. This contradicts with the ionic model which predicts a stronger tilting instability with a smaller tolerance factor.

The above statement is supported by our projected AIMD analysis at 80 K (Fig. 3(c) and (d)). First, at low frequency (<30 cm⁻¹), the tilting motion in CsSnBr₃ is dominant (Fig. 3(d)). Its weight continuously decays with increasing frequency, and corresponds to the fitted Debye component in the Raman spectrum (Fig. 3(b)). This is in contrast to what is observed for CsPbBr₃ (Fig. 3(c)), where the relaxational behavior of the tilting motion is not as significant as in CsSnBr₃, and the low frequency dynamics are also shared by Cs and Br₆ distortion motions. The peak at 30 cm⁻¹ in both CsSnBr₃ and CsPbBr₃ can be ascribed to Cs motion. The peak at 50 cm⁻¹ in CsPbBr₃ is also assigned to distortion of the octahedra. In CsSnBr₃, distortion motions cover a wider frequency range and likely contribute to the Raman peaks at 40 cm⁻¹–80 cm⁻¹. Finally the features between 70 cm⁻¹–90 cm⁻¹ in CsPbBr₃ are connected with stretching and M cation motion.

Similar to the tilting instability of the cubic CsGeBr₃ discussed previously, the observed Raman spectra and the results of the projected AIMD analysis for the orthorhombic phases cannot be rationalized by the geometric model that predicts stronger tilting instability for CsPbBr₃ (t = 0.86) compared to CsSnBr₃ (t = 0.92). This suggests that the stronger tendency for lone-pair driven distortions for Sn²⁺ contributes to the tilting motion, in concert with intra-octahedral distortions at higher frequencies (green lines in Fig. 3(c) and (d)).

We note that there is additional evidence that supports our suggestion that CsSnBr₃ exhibits an instability even at 80 K in the orthorhombic phase. Fig. S4 in the ESI† presents the temperature evolution of the Raman spectrum in the 10 K–80 K range. Around 60 K, we find indication of another phase transition to lower symmetry (possibly monoclinic²³) phase. This phase transition is manifested by the appearance of several new peaks and some changes in the existing modes. Furthermore, Fig. S5 in the ESI† presents 0 K, DFT-based phonon dispersion relations that show imaginary modes for orthorhombic CsSnBr₃ but not for orthorhombic CsPbBr₃.

Conclusions

Our work illustrates the critical role of the metal cation s lone pairs on the nature of the structural dynamics and represents a first step towards tuning the dynamics of perovskite structures. We investigated how the structural dynamics of CsMBr₃ (M = Pb, Sn, and Ge) are affected by the stereochemical electron lone pairs on the metal cations. By simultaneously decreasing the energy scale of the tilting instability and increasing the energy scale of the lone-pair polar distortion across the series CsPbBr₃ → CsSnBr₃ → CsGeBr₃, our primary finding is that for all three compounds, the lone-pair stereochemical activity contributes significantly to octahedral tilting instabilities that are unaccounted for in the geometric models or DFT computations at 0 K. This strong instability leads to liquid-like (i.e. central peak) Raman activity in the cubic phase of all three crystals. Since the octahedral tilt angles directly control carrier transport, this study provides the crucial link between chemical composition and optoelectronic properties, opening opportunities for lead-free solution-processable photovoltaics.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

O. Y. acknowledges funding from BSF (grant No. 2016650) and ERC (850041–ANHARMONIC). D. H. F. gratefully acknowledges financial support from the Alexander von Humboldt Foundation. M. G. K. acknowledges support form the Department of Energy, Office of Science, Basic Energy Sciences, under Grant No. SC0012541 (sample synthesis, structure and property characterization). L. G. was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award # DE-FG02-07ER46431. Computational support was provided by the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy, Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231. A.M.R. acknowledges the support of the Office of Naval Research, under grant number N00014-20-1-2701.

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d1ma00288k

‡ These authors contributed equally.

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