Hybrid Perovskite at Full Tilt: Structure and Symmetry Relations of the Incommensurately Modulated Phase of Methylammonium Lead Bromide, MAPbBr 3

: As energy-conversion materials, organic − inorganic hybrid perovskites remain a research-and ﬁ nance-intensive topic. However, even for the arguably most iconic representatives, methylammonium and formamidinium lead halides, the crystal structures of several polymorphs have remained undetermined. Herein, we describe the incommensurately modulated structure of MAPbBr 3 in (3+1)D superspace, as deduced from single-crystal X-ray di ﬀ ractometry despite systematic twinning. A ﬃ rming the published average space group, we determined the superspace group Imma (00 γ ) s 00 with cell parameters of a = 8.4657(9), b = 11.7303(12), c = 8.2388(8) Å, and q = 0.2022(8) c * . Via group − subgroup and mode analyses using irreducible representations, we establish symmetry relationships to the well-known cubic and orthorhombic polymorphs and break down distortions into the average tilt system a − b 0 a − and modulated contributions to tilt and deformation of the PbBr 6 coordination polyhedra. Not only does our model ﬁ ll a long-standing gap in structural knowledge, but it may also serve as a starting point for elucidating other modulated structures within this substance class.

I n the field of energy-conversion materials, organic− inorganic hybrid perovskites are a persistent hot topic that has given rise to extensive research, deduction, prognostication, speculation, and any mix thereof. 1,2And while derived perovskite solar cells (PSCs) are already being prototyped and announced for commercialization, many questions concerning underlying crystal-structural principles remain within this surprisingly complex substance class.
Organic−inorganic hybrid perovskites, in the strict sense of the word, 3 share the formula ABX 3 , with A being a molecular organic cation (typically methylammonium, MA, or formamidinium, FA), B a metal cation (often lead or tin), and X an anion (mostly halide)although mixed occupation may occur at any position.Most of these compounds come in three perovskite-homeotypic modifications: the cubic high-temperature α phase, the tetragonal medium-temperature β phase, and the orthorhombic low-temperature γ phase.The crystal structures of these have been characterized with respect to the orientation/position of the organic cation, 4 their actual spacegroup type, 5 and their symmetry relationshipsand not all facts go undisputed.Besides the rather well-known polymorphs, two further classes of modifications have been described: different δ phases with non-perovskite structures (e.g., for FAPbI 3 ) 6 and phases with incommensurately modulated structures, generally dubbed "the incommensurate (IC) phase".The latter occurs at the fringe of the thermally induced β⇄γ transformation, most prominently of MAPbCl 3 , 7 but has also been reported for methylammonium lead(II) tribromide (MAPbBr 3 ) and FA congeners. 8,9 early as in 1987, an intermediate MAPbBr 3 phase (then called "γ") at 149.5−155.1 K was reported with a structure conforming to the space group P4/mmm, a = 5.894(2) and c = 5.861(2) Å, with Z = 1, as evidenced by indexed Guinier− Simon patterns. 10Later calorimetric measurements suggested a stability range of 148.8−154.0K, 11 before a superstructure in the space-group type I4/mmm with Z = 4 was suggested based on single-crystal diffractometry. 7Incommensurability was postulated in analogy to the chloride congener 12 but was only thoroughly described by Guo et al. in 2017. 13Based on single-crystal X-ray diffraction, they assigned the space-group type Imma (Z = 4) to the average structure and estimated a temperature-dependent modulation vector q = γc* (referring to the standard orthorhombic setting) with 0.2073 ≥ γ ≥ 0.1773 in the range of 148−155 K. Furthermore, the twinning that accompanies the transformations β⇄IC⇄γ was also addressed, but it probably kept the authors from solving the atomic structure of the IC phases.We were able to overcome this problem and present, for the first time, the incommensurately modulated crystal structure of MAPbBr 3 at 150 K, where satellite reflections are most pronounced, in the modern (3+1)D superspace description. 14ur investigation is based on a data set previously collected for another study with a different focus. 13(The respective crystal had been grown from an aqueous dimethylformamide solution of dimethylamine, hydrogen bromide, and lead(II) iodide using the antisolvent method with 2-propanol as precipitant; see Supplementary Information of ref 13 for details.)Most software packages for data reduction do not support simultaneous handling of twinning and modulation.We were, however, successful in only integrating reflections caused by the major twin domain and dealing with partial overlap during refinement with JANA2006/2020. 15The twin law is a rotation of almost 180°ca.around [101] (i.e., swapping of a and c, inversion of b), which is lost as a symmetry operation during the tetragonal-to-orthorhombic phase transition.The structure conforms to the (3+1)D superspace group Imma(00γ)s00, a = 8.4657 (9), b = 11.7303(12), and c = 8.2388(8) Å, q = 0.2022(8)c*, with Z = 4, leading to an average structure in the space-group type Imma, as described by Guo et al. before. 13Although q equals the rational fraction of 1 / 5 c* within 3σ, the authors have shown the modulation vector to vary with temperature.Therefore, we refrain from suggesting a commensurate model but deem a 1×1×5 cell a suitable approximant (cf.model with starting phase t 0 = 0 in the Supporting Information (SI)).The [ ] − ∞ PbBr 3 3 framework occupies one unique lead and two unique bromide positions (for bond lengths, see Table S2 in the SI).The MA ion was found in two unique perpendicular orientationsone with the C−N bond along b, one with it almost along ain a ratio of 0.33(3):0.67(3).Each of them is disordered over a mirror plane perpendicular to the bond, so that carbon and nitrogen positions are interchangeable.Because only first-order satellite reflections were discernible, modulations (as aforementioned plus displacement modulation for lead and bromide ions) were modeled with harmonic functions of first order.Discontinuous models proved unwarranted, despite slightly larger absolute residual density (cf.Figures S1−S6 in the SI).Compared to routine structure analyses, the quality of the final reduced data (R σ = 3.52%, R int = 12.09%) and the refined model (R 1 = 8.16%, wR 2 = 13.13%, and S = 1.4929 for all data) suffers from the somewhat approximate treatment of twinning as well as the overall low intensity and overlap of satellite reflections.Keeping these systematic problems in mind, the data and model nonetheless provide a sound basis for the interpretation of the crystal structure and its symmetry relationships (see section 1 in the SI for details on structure solution, refinement, and final model).
The nature of the structural modulation is best understood referring to the average structure (see Figure 1d).Besides minor contributions, the latter is related to the cubic perovskite aristotype structure by two equally large out-of-phase tilts of the PbBr 6 coordination octahedra along two Pb−Br−Pb axes (tilt system: a − b 0 a − ).This distortion effects a descent from the space group Pm3̅ m (a, b, c) to the non-maximal subgroup Imma (a + b, 2c, a − b; see Figure 2).Positional modulation is relatively weak for the lead but strong for the bromide ions.Treatment of the MA ion as a rigid molecule showed that the actual position near 0, 1, 1 / 2 is governed by hydrogen-bonding to the bromide ions (at least two NH•••Br bonds per orientation, criteria: S3 in the SI).In this way, the positional modulation of the bromide ions is linked to the translational/ rotational modulation of the organic cations.Because of the comparatively low X-ray scattering power and disorder over a position of higher symmetry, we refrain from discussing the MA positions/orientation in more depth and will thus focus on the [ ] − ∞ PbBr 3 3 framework.The incommensurate modulation of bromide positions mainly affects the tilts of distorted PbBr 6 octahedra: While, on average, they conform to a − in two directions (see above), they vary harmonically between no tilt and a maximal out-of-phase tilt, phase-shifted by half a period with respect to each other (see Figure 3 and t movie in the SI).A slight twist of the basal Pb(Br1) 4 plane, which is tilted with respect to the ac plane, occurs concomitantly.One modulation period spans 1/γ cells in the c direction, so that 2γ changes per cell between zero and maximum tilt ensue.As γ increases with decreasing temperature, 13 so does the number of changes in a given crystal segment.For γ→1 or γ→0, the a − b + a − structure of low-temperature MAPbBr 3 or a hypothetical a − b 0 c − structure in C2/m (−a − c, −b, c) would be approached, respectively.
The bromide displacement parameters at each modulation phase reflect the bonding, tilt, and twist situation in a physically sensible manner.While the displacement modulation of Pb1 is very small but significant, Br1 and Br2 exhibit only one significant and large modulation parameter each: the ones of U 22 (Br1) and U 13 (Br2) (cf.t movie in the SI).This means that, for Br1, the magnitude of displacement along b increases with bending the (ideally linear) Pb−Br−Pb Atoms are represented as ellipsoids/spheres of 50% probability (gray, lead; brown, bromine; black, carbon; blue, nitrogen; white, hydrogen) as drawn using VESTA. 16e Journal of Physical Chemistry Letters arrangement along b, signifying enlarged anisotropic vibrational motion.For Br2, on the other hand, the modulation of displacement orientation in the ac plane conveys a dynamic shift toward the nearest hydrogen-bonding donor at each phase.
Analysis of the structural distortion modes induced by irreducible representations of space-group symmetries using ISODISTORT gives further insight into the phase relationships α→IC→γ, especially with respect to PbBr 6 tilts and distortions. 18,19The main idea of this method is to consider the structural degrees of freedom not in terms of individual atomic coordinates but of collective displacements (the socalled displacive distortion modes) that transform according to the irreducible representations of the higher-symmetry parent space group.(We will not consider strain modes, which convey the rather trivial changes of unit-cell lengths.)For this purpose, we have separated the hypothetical transformation of the cubic α-MAPbBr 3 aristotype into the average IC-MAPbBr 3 structure from the incommensurate distortion into the actual IC-MAPbBr 3 hettotype (see Figure 4 and the detailed breakdown in Table S4 in the SI).The positional parameters from the structural models were then decomposed into distortion modes and assigned to specific changes in the PbBr 6 polyhedron after inspection.Because of its disorder, the MA ion was excluded by placing an invariant pseudoatom in an idealized position.For each transformation, one single major distortion mode was found (transforming like R 4 + , Λ 4 , and X 4 + , respectively) and identified as "primary", which means it can effect the symmetry lowering on its own.For α→IC (average), this conveys the octahedral a − b 0 a − tilt as described above.Within the orthorhombic system, the PbBr 6 environment does not have a strictly octahedral shape and is subject to distortion with respect to an equatorial base (ca. in the ac plane) and an apical axis (ca.along b).A shear of the base and a tilt of the axis, each in two directions, lead to the well-known γ-MAPbBr 3 in the maximal subgroup type Pnma (thus, this transformation may be of second order according to Landau theory).In the gedankenexperiment of following IC (average)→IC, the same incommensurate distortion manifests as tilt or twist / shear or shift, depending on the position in the crystal.Overall, the modulation causes an additional tilt of the PbBr 6 polyhedron with concomitant deformation of the base.The reason for the occurrence of incommensurability within a narrow temper-   c, d) corresponding to two main axes of the cubic perovskite aristotype.Atoms are drawn with arbitrary radii (gray, lead; brown, bromine) using VESTA; 16 MA ions were omitted for clarity.The average structure exhibits the same a − out-of-phase tilt in both directions, while the modulated structure varies from almost zero (green circle) to nearly maximal out-of-phase tilt (red circle).Please note that the modulation direction is none of the bisectors but c||c* (cf. Figure 1a−c).
The Journal of Physical Chemistry Letters ature range seems to be structural frustration during the reconstructive first-order transition β→IC, which crosses over from another branch of the Barnighausen tree.This is evidenced by some locally rather unfavorable deformations (globally thinned out via modulation) that relax into the periodic pattern of γ-MAPbBr 3 upon further cooling.
For our study, we have solved and refined the hitherto unknown incommensurately modulated structure of MAPbBr 3 at 150 K and presented a model of ample quality despite systematic problems.(Crystals suffer from twinning caused by the symmetry descent from P4/mmm (a, b, c) to Cmmm (a − b, a + b, c) underlying the reconstructive β→IC transformation.)The average orthorhombic unit cell in the spacegroup type Imma gives rise to a known perovskite variant derived by an a − b 0 a − tilt from the cubic aristotype. 20With a ≈ 8, b ≈ 12, and c ≈ 8 Å, and Z = 4, the parameters are typical for the system.The actual structure in the (3+1)D superspace group Imma(00γ)s00, q ≈ 0.2c*, is subject to additional distortions: Via mode analysis, these were identified to transform like the irreducible representation Λ 4 and manifest in an additional incommensurate tilt of the PbBr 6 polyhedra and deformations of their equatorial plane.We tentatively assign this to structural frustration induced during the firstorder β→IC transformation, which is resolved through restoration of the periodicity in the IC→γ transformation upon further cooling.Not only do our model and the in-depth analysis of its symmetry relationships shed light on one of the most iconic organic−inorganic hybrid perovskites, but they may also serve as a starting point for pinning down and elucidating other (suspected) modulated structures within this substance class (e.g., of MAPbCl 3 , 7,21 MAPbI 3 , 22 FAPbBr 3 , 8 or FAPbI 3 ).
Movie showing the 3D crystal structure as a function of the cell modulation phase t, hydrogen atoms with arbitrary radius, all other atoms as ellipsoids of 50% probability (gray: Pb, brown: Br, black: C, blue: N; white: H; Br1 and Br2 in ac plane at y ≈ 0 and 1 / 2 or at y ≈ 1 and 3 / 4 , respectively), MA ions in both unique orientations with arbitrary C/N assignment (omitted at ca. 1 / 2 , 1, 1 and 1 / 2 , 3 / 4 , 1 for clarity), cell borders in gray (MP4) Experimental and geometrical details, de-Wolff sections, and details on symmetry relationships (PDF) Basic crystallographic information file for a 2×1×5 approximant cell with the starting phase t 0 = 0, caveat: aperiodic beyond boundaries (CIF)  The Journal of Physical Chemistry Letters

Figure 1 .
Figure 1.Section from the crystal structure of IC-MAPbBr 3 (approximant with origin at phase t 0 = 0, aperiodic along c) viewed roughly along (a) a, (b) −b, and (c) −c; hydrogen atoms omitted for clarity, MA ions shown in both unique orientations with arbitrary carbon/nitrogen assignment.(d) Average crystal structure of IC-MAPbBr 3 viewed roughly along a; MA ions only shown in one arbitrary unique orientation for each position with arbitrary carbon/ nitrogen assignment.Atoms are represented as ellipsoids/spheres of 50% probability (gray, lead; brown, bromine; black, carbon; blue, nitrogen; white, hydrogen) as drawn using VESTA.16

Figure 2 .
Figure 2. Barnighausen tree showing the group−subgroup relationships between the discussed structures (discussed path in black, alternative in gray).Nodes consist of full Hermann−Mauguin symbols for the space-group type, tilt systems in Glazer notation (if a representative exists), 17 occupied Wyckoff positions (by A + , Pb 2+ , and Br − ions), and a phase representative (if existent, with dark background).Links are annotated with kind and index of relationship as well as transformation of the cell basis (if applicable).

Figure 3 .
Figure 3.Sections of the average (a, c) and modulated crystal structure at t 0 = 0 (b, d) viewed roughly along a + c (a, b) and a − c (c, d) corresponding to two main axes of the cubic perovskite aristotype.Atoms are drawn with arbitrary radii (gray, lead; brown, bromine) using VESTA;16 MA ions were omitted for clarity.The average structure exhibits the same a − out-of-phase tilt in both directions, while the modulated structure varies from almost zero (green circle) to nearly maximal out-of-phase tilt (red circle).Please note that the modulation direction is none of the bisectors but c||c* (cf.Figure1a−c).

Figure 4 .
Figure 4. Distortion modes describing the (hypothetical) transformations between different polymorphs of MAPbBr 3 .Nodes name the polymorph and its (super)space-group type; links are annotated with coordinate transformations as well as irreducible representations, affected atoms, and description of the predominant displacive distortions."Base" refers to the Pb(Br1) 4 coordination plane, "axis" to the Br2−Pb−Br2 arrangement.