Twinning in MAPbI3 at room temperature uncovered through Laue neutron diffraction

The crystal structure of MAPbI3, the signature compound of the hybrid halide perovskites, at room temperature has been a reason for debate and confusion in the past. Part of this confusion may be due to twinning as the material bears a phase transition just above room temperature, which follows a direct group–subgroup relationship and is prone to twinning. Using neutron Laue diffraction, we illustrate the nature of twinning in the room temperature structure of MAPbI3 and explain its origins from a group-theoretical point-of-view.

www.nature.com/scientificreports/ temperature and that was never heated above the cubic-tetragonal phase transition temperature. This study is seconded by an in-depth explanation of twinning in MAPbI 3 in order to make the thematic approachable for the wider scientific community. twinning in MApbi 3 through the cubic-to-tetragonal phase transition. In a nutshell, twinned crystals are not composed of one single orientation, but are built up from different individual or domains. These domains or individual are, however, not randomly oriented but related through point symmetry elements, which must not be element of the space group symmetry, and can be categorised through the twinning element in inversion twins (related through inversion at a twin inversion centre), rotation twins (related through rotation at a twin axis) or reflection twins (related through reflection at a twin plane) 31 . For twinning that occurs during a phase transition, the twinning element needs to be a symmetry element of the high-symmetry structure, which is lost during the phase transition. Therefore, it is crucial to understand the group-subgroup relationships through the cubic-to-tetragonal phase transition in MAPbI 3 . Prior work on twinning in the perovskite BaTiO 3 may be taken as a first guidance 32 . The phase transition from the cubic perovskite aristotype in Pm3m to the room temperature perovskite structure in I4cm (I4/mcm) follows a direct group-subgroup relationship and hence makes a second order phase transition possible. Even more importantly, the t3 translationengleiche symmetry descent from Pm3m to P4/mmm ( Fig. 1) gives rise to the formation of triple twinning along the lost symmetry elements, i.e. the four threefold rotation axes. It is probably more illustrative to approach this from the unit cell dimensions (which are, of course, a consequence of the symmetry operations): while all three unit-cell axes are constrained to be equal in the cubic crystal system, only a and b have to fulfil this constraint in the tetragonal crystal system with c being independent of the other two. When transitioning from cubic to tetragonal, the choice of which of the three cubic unit-cell axes becomes the independent c-axis is arbitrary and hence three equivalent possibilities exist, standing perpendicular to each other.
A complication of the cubic-to-tetragonal transition in MAPbI 3 is the symmetry descent from P4/mmm into the body-centred klassengleiche subgroup I4cm (I4/mcm; Fig. 1). This transition is accompanied with a 45° rotation of the unit cell in the ab-plane and thus a change of the lattice constants in the manner a t−I = √2•a c and c t−I = 2•c c (where the index t − I signifies the tetragonal body centred structure and c the cubic one). This has as consequence that direction and plane denominations change in a rather complex manner when transforming from Pm3m to I4cm (I4/mcm). Therefore, we chose a non-standard setting of the tetragonal unit cell in F4mc (F4/mmc), where a t−F = 2•a c and c t−F = 2•c c (with t − F standing for tetragonal face centred; Fig. 1). Although this setting violates the Bravais rules of choosing the smallest possible unit cell, it has as major advantage that the directions correspond to each other in the tetragonal and the cubic lattices. Where appropriate, we will give directions and lattice planes for both settings. At room temperature, the lattice constants a and c with regard to the non-standard setting F4mc (F4/mmc) are not greatly different (a t−F = 12.507 Å, c t−F = 12.622 Å) 21 , which means that the diffraction spots of the different domains are close to each other to the extent that they nearly overlap completely-similar to a pseudo-merohedral twinning.
The twinning element in the cubic-to-tetragonal descent can, essentially, be any symmetry element lost during the phase transition. Most prominently, the aforementioned three-fold axes, which are defining the cubic crystal system, can become the twinning element. However, rotation twinning through 90° rotations along the <100> c axes or 180° rotation along the <101> c axes (Fig. 2) are also possible as well as reflection twinning on the {011} c -planes [23][24][25] . These twin laws correspond to the respective four-fold and two-fold rotation symmetries that are lost when c becomes unequal from a and b in the transition from the cubic aristotype to the tetragonal www.nature.com/scientificreports/ structure. However, these different twin laws are only seemingly contradictory, as they all represent the same phenomenon of axis permutations, i.e. of mapping the unique c axis on either a or b axis (Fig. 2). The difference between the different twin laws is the relative orientation of the a and b axes, which are symmetry equivalent in the tetragonal crystal system and are linked through four-fold rotation along the [001] axis and {100} mirror planes. Furthermore, a non-centrosymmetric space group would only be visible in the reflection intensities, not in their positions. A 120° rotation around one of the <111> c directions would hence produce twins that look identical to 90° twins around the <100> c directions or 180° twins around the <101> c directions. The same also www.nature.com/scientificreports/ applies to the {011} c reflection twins, that have been reported for this system. Finally, it needs to be noted that the presence of these twin laws does not rule the possibility of further inversion twinning out and, therefore, does not allow a conclusion on the absence of ferroelectricity: Inversion twinning would only be discernible under very specific measurement conditions, for instance at the proximity of an element's absorption edge, since Friedel's law is not strictly valid for these cases. Two structural effects occur during the cubic-to-tetragonal phase transition within the crystal structure of MAPbI 3 : (1) the molecular cations are no longer completely disordered, but orient along a number of preferred orientations and (2) a tilting of the [PbI 6 ] octahedra, both of which phenomena are probably interlinked. (1) The molecular cation in the cubic crystal structure is completely disordered and rotates freely in its cuboctahedral void 33 . In the room temperature crystal structure, though, it is still dynamically and statistically disordered but resides in well-defined preferential orientations in correspondence to its surrounding anions 9 . (2) The network of corner-sharing octahedra is undistorted in the cubic aristotype, i.e. all Pb-I-Pb angles are at 180° as dictated by the cubic symmetry. At room temperature, however, the octahedra are still untilted in one direction, i.e. the Pb-I-Pb angles along the crystallographic c-axis remain at 180°, but they are tilted within the ab-plane. From a crystallographic point of view, the PbI 6 units no longer form exact octahedra in the tetragonal symmetry, but rather a square-planar bipyramid. The tilting of the inorganic [PbI 6 ] octahedra affects the size of the A-cation void (the place of the molecular MA + cation) and on the electronic structure of the compound, which is mainly determined by the inorganic [PbI 6 ] octahedra network.
Neutron Laue diffraction on MAPbI 3 . When applying diffraction methods, one is faced with two distinct problems: on the one hand powder diffraction techniques do not allow to distinguish between I4/mcm and I4cm as both belong to the same diffraction group 34 , i.e. they bear exactly the same translation symmetry and hence have the exact same systematic extinctions. Therefore, the only reflections that could differ overlap entirely. Single crystal studies, on the other hand, are flawed by heavy twinning of the crystals under consideration 35 , which is a consequence of the phase behaviour of this compound: MAPbI 3 shows several structural phase transitions as outlined above.
Applying neutron Laue diffraction to study twinning in methylammonium lead iodide has two major advantages: first, neutrons interact much less strongly with matter than X-rays as neutrons are scattered on the atomic nuclei only, while X-rays are scattered on the electron shell. This implies that larger samples are necessary for neutron diffraction, but also less absorption for the electron rich atoms, such as lead and iodine. The larger size of the samples in neutron diffraction is indeed an advantage for this study, as these crystals typically exhibit more pronounced twinning. Furthermore, the architecture of the instrument is adapted to the use of larger crystals, which is not typically the case for X-ray diffraction with normally much smaller beam sizes of ≈ 500 µm. The chosen setup allowed us to measure crystals of several millimetres size. Furthermore, Laue diffraction has a unique property: by using a polychromatic ("pink") beam, each individual wavelength constructs its own Ewald sphere and, therefore, allows measurements with static detectors. In a typical single wavelength experiment-which is the standard in X-ray diffraction-the reflections move in and out of the detector plane (i.e. the plane in which the detector records) when the lattice constants change, for instance due to temperature. Therefore, it is generally necessary to measure a crystal during a rotation. In Laue diffraction, on the other hand, the polychromatic light produces reflection spots regardless of lattice parameter changes, and a crystal rotation is not necessary. Using powder diffraction does not allow the distinction of individual domains, since the three-dimensional information of the reflection position is merged into a single dimension, the scattering angle. However, the twinned reflections appear, because more than one domain overlay in a very specific way that is only discernible in three dimensions.
Due to the heavy twinning of these crystals, it was impossible to index the neutron Laue patterns using the unit cell parameters of the tetragonal room temperature crystal structure in the space group I4cm (I4/mcm) directly. However, indexation was successful on basis of the cubic aristotype, as the differences in the reflection positions between the cubic aristotype and the tetragonal room-temperature structure are rather small. The hkl indices of the cubic indexing are given in Fig. 3, and can be linked to the tetragonal ones through the groupsubgroup relationship between the two space groups (comp. Fig. 1). To distinguish the hkl indices in this work, tetragonal body centred, tetragonal face centred and cubic hkls indices are marked with an index t − I, t − F or c, respectively. Furthermore, we checked the successful indexation using the tetragonal subgroup P4/mmm (see "SI"), which is an intermediate space group in the symmetry descent but still bears approximately the same lattice constants as the cubic aristotype. In essence all three axis permutations appear as equally acceptable solutions for the indexation.
Using a static crystal in a special sample environment, we conducted temperature-dependent measurements using a specific crystal orientation. Due to the sample environment, we were limited in the choice of orientation, which is why the orientations at room temperature and at controlled temperature differ. The Laue patterns clearly show a twinning at room temperature in a pseudo-merohedral way: the double-maxima of the spots under consideration lie very close to each other (Fig. 4a). However, the Laue spots are clearly split at 300 K (Fig. 4b,c). On heating towards the tetragonal-to-cubic phase transition temperature at 330 K, the spots merge more and more into one single spot as the reflections of the twins become equivalent and hence the twinning domains are being merged during this process. The smooth merging of the domains can be taken as an indicator for the fact that this phase transition is actually an order-disorder phase transition.
In order to prove the nature of the twinning at room temperature, we recorded a series of diffraction images while turning the crystal in increments of 1° around the ω-axis (i.e. the axis, on which the crystal is mounted and that is perpendicular to the beam path) without the heating equipment. What is observed in such an experiment is the movement of the diffraction spots through the detector plane, caused by diffraction at the same lattice plane from different wavelengths at different crystal rotations. According to Bragg's law, the separation of reflections Consequently, the 111 c reflection does not exhibit a change in splitting over its full way through the detector plane, in line with rotation twinning along the three-fold <111> axes. As we grew the crystals used in this experiment at room temperature, the twinning clearly does not stem from running through the cubic-to-tetragonal phase transition. Instead, a different symmetry between the crystallisation nuclei and the bulk material may be an explanation for this behaviour, where the nuclei have a higher symmetry than the bulk material.

conclusion
We demonstrated the particularities of twinning in MAPbI 3 by means of neutron Laue diffraction and explained the peculiarities of twinning in MAPbI 3 from a crystallographic point-of-view. We could show intrinsic twinning in crystals grown at room temperature, which have not been heated to the existence region of the cubic phase prior to the experiments. This latter gives a powerful insight into the crystallisation and growth process of MAPbI 3 and supports the assumption of cubic crystallisation nuclei. This finding is important as it may allow more targeted thin film production procedures in the future, and more targeted studies on the crystallographic nature of the earlier crystallisation states will be highly beneficial.

Materials and methods
Single crystals for this study were grown using an antisolvent room temperature method described before 16,21 . Crystals were covered in acrylic lacquer to prevent oxygen and moisture destruction of the crystal. Neutron Laue patterns were collected using the FALCON (Fast Acquisition Laue Camera for Neutrons) Laue diffractometer at BER II research reactor (HZB). FALCON consists of two area detectors in backscattering and transmission position with four iCDD cameras each and total scattering area of 400 × 400 mm 2 . A "pink" neutron beam with wavelength ≈ 0.9-3.2 Å and a neutron flux of 8 × 10 6 n/cm 2 s is applied. For variable temperature measurements, a detector distance for the Laue patterns was 168 mm, pattern acquisition time was 60 s. Temperature depending measurements were carried out by means of a Stirling cooler based closed cycle cryostat providing sample temperatures from 150 up to 450 K 36 . Room temperature measurements were performed at a detector distance of 120 mm with an acquisition time of 180 s. By means of a rotation stage, ω-scans were performed within the range of − 120° and 120°. For indexing of the Laue patterns "Cologne Laue Indexation programme" CLIP 37 was applied. Successful indexation was possible assuming perovskite cubic aristotype with a unit cell dimension of a = 6.3 Å. Further, indexation was performed in the tetragonal subgroup P4/mmm, as this subgroup