Hybrid Perovskite at Full Tilt: Structure and Symmetry Relations of the Incommensurately Modulated Phase of Methylammonium Lead Bromide, MAPbBr3 [original]

Hybrid Perovskite at Full Tilt: Structure and Symmetry Relations of

the Incommensurately Modulated Phase of Methylammonium Lead

Bromide, MAPbBr3

Dennis Wiedemann,*Joachim Breternitz,*Daniel W. Paley, and Susan Schorr

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sıSupporting Information

ABSTRACT: As energy-conversion materials, organic−inorganic hybrid perovskites remain a

research- and ﬁnance-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 MAPbBr3in (3+1)D superspace, as deduced from single-crystal X-ray

diﬀractometry despite systematic twinning. Aﬃrming the published average space group, we

determined the superspace group Imma(00γ)s00 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−b0a−

and modulated contributions to tilt and deformation of the PbBr6coordination polyhedra. Not

only does our model ﬁll 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.

Intheﬁeld 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.

1,2

And 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,

share the formula ABX3, with Abeing a molecular

organic cation (typically methylammonium, MA, or formamid-

inium, FA), Ba metal cation (often lead or tin), and Xan

anion (mostly halide)although mixed occupation may occur

at any position. Most of these compounds come in three

perovskite-homeotypic modiﬁcations: the cubic high-temper-

ature αphase, the tetragonal medium-temperature βphase,

and the orthorhombic low-temperature γphase. The crystal

structures of these have been characterized with respect to the

orientation/position of the organic cation,

their actual space-

group type,

and their symmetry relationshipsand not all

facts go undisputed. Besides the rather well-known poly-

morphs, two further classes of modiﬁcations have been

described: diﬀerent δphases with non-perovskite structures

(e.g., for FAPbI3)

and phases with incommensurately

modulated structures, generally dubbed “the incommensurate

(IC) phase”. The latter occurs at the fringe of the thermally

induced β⇄γtransformation, most prominently of MAPbCl3,

but has also been reported for methylammonium lead(II)

tribromide (MAPbBr3) and FA congeners.

8,9

As early as in 1987, an intermediate MAPbBr3phase (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.

Later calorimetric measurements suggested a

stability range of 148.8−154.0 K,

before a superstructure in

the space-group type I4/mmm with Z= 4 was suggested based

on single-crystal diﬀractometry.

Incommensurability was

postulated in analogy to the chloride congener

but was

only thoroughly described by Guo et al. in 2017.

Based 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⇄γwas 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 incommensur-

ately modulated crystal structure of MAPbBr3at 150 K, where

satellite reﬂections are most pronounced, in the modern

(3+1)D superspace description.

Received: December 18, 2020

Accepted: January 21, 2021

Published: March 5, 2021

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Our investigation is based on a data set previously collected

for another study with a diﬀerent focus.

(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.

The twin

law is a rotation of almost 180°ca. around [101] (i.e.,

swapping of aand 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.

Although qequals

the rational fraction of 1/5c*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 t0= 0 in the Supporting Information

(SI)). The []

−

∞PbBr3

3framework 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 modula-

tion 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 (Rσ= 3.52%, Rint

= 12.09%) and the reﬁned model (R1= 8.16%, wR2= 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 PbBr6coordination octahedra along two Pb−Br−Pb axes

(tilt system: a−b0a−). This distortion eﬀects a descent from the

space group Pm3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, 1/2is governed by hydrogen-bonding

to the bromide ions (at least two NH···Br bonds per

orientation, criteria: d(N···Br) < 3.8 Å and ∠

(N−H···Br) ≥

140°; see Table 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 []

−

∞PbBr3

3framework. The incommensurate modulation of

bromide positions mainly aﬀects the tilts of distorted PbBr6

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 tmovie in the SI).

A slight twist of the basal Pb(Br1)4plane, which is tilted with

respect to the ac plane, occurs concomitantly. One modulation

period spans 1/γcells in the cdirection, so that 2γchanges per

cell between zero and maximum tilt ensue. As γincreases with

decreasing temperature,

so does the number of changes in a

given crystal segment. For γ→1orγ→0, the a−b+a−structure

of low-temperature MAPbBr3or a hypothetical a−b0c−

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 modu-

lation of Pb1 is very small but signiﬁcant, Br1 and Br2 exhibit

only one signiﬁcant and large modulation parameter each: the

ones of U22(Br1) and U13(Br2) (cf. tmovie in the SI). This

means that, for Br1, the magnitude of displacement along b

increases with bending the (ideally linear) Pb−Br−Pb

Figure 1. Section from the crystal structure of IC-MAPbBr3

(approximant with origin at phase t0= 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-

MAPbBr3viewed 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.

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J. Phys. Chem. Lett. 2021, 12, 2358−2362

2359

arrangement along b, signifying enlarged anisotropic vibra-

tional 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 relation-

ships α→IC→γ, especially with respect to PbBr6tilts and

distortions.

18,19

The 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 so-

called 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-MAPbBr3aristotype into the average IC-MAPbBr3structure

from the incommensurate distortion into the actual IC-

MAPbBr3hettotype (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 speciﬁcchangesinthePbBr

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 R4+,Λ4, and X4+,

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−b0a−tilt as described above.

Within the orthorhombic system, the PbBr6environment 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-MAPbBr3

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 PbBr6polyhedron

with concomitant deformation of the base. The reason for the

occurrence of incommensurability within a narrow temper-

Figure 2. Barnighausen tree showing the group−subgroup relation-

ships 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),

occupied Wyckoﬀpositions (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. Sections of the average (a, c) and modulated crystal

structure at t0= 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;

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).

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J. Phys. Chem. Lett. 2021, 12, 2358−2362

2360

ature range seems to be structural frustration during the

reconstructive ﬁrst-order transition β→IC, which crosses over

from another branch of the Barnighausen tree. This is

evidenced by some locally rather unfavorable deformations

(globally thinned out via modulation) that relax into the

periodic pattern of γ-MAPbBr3upon further cooling.

For our study, we have solved and reﬁned the hitherto

unknown incommensurately modulated structure of MAPbBr3

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)toCmmm

(a−b,a+b,c) underlying the reconstructive β→IC trans-

formation.) The average orthorhombic unit cell in the space-

group type Imma gives rise to a known perovskite variant

derived by an a−b0a−tilt from the cubic aristotype.

With 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 Λ4and manifest

in an additional incommensurate tilt of the PbBr6polyhedra

and deformations of their equatorial plane. We tentatively

assign this to structural frustration induced during the ﬁrst-

order β→IC transformation, which is resolved through

restoration of the periodicity in the IC→γtransformation

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 MAPbCl3,

7,21

MAPbI3,

FAPbBr3,

FAPbI3).

■ASSOCIATED CONTENT

sıSupporting Information

The Supporting Information is available free of charge at

https://pubs.acs.org/doi/10.1021/acs.jpclett.0c03722.

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 1/2or 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-Wolﬀsections,

and details on symmetry relationships (PDF)

Basic crystallographic information ﬁle for a 2×1×5

approximant cell with the starting phase t0= 0, caveat:

aperiodic beyond boundaries (CIF)

■AUTHOR INFORMATION

Corresponding Authors

Dennis Wiedemann −Department Structure and Dynamics of

Energy Materials, Helmholtz-Zentrum Berlin fur Materialien

und Energie GmbH, 14109 Berlin, Germany; orcid.org/

0000-0001-6294-3205; Email: dennis.wiedemann@

helmholtz-berlin.de

Joachim Breternitz −Department Structure and Dynamics of

Energy Materials, Helmholtz-Zentrum Berlin fur Materialien

und Energie GmbH, 14109 Berlin, Germany; orcid.org/

0000-0002-0192-6919; Email: joachim.breternitz@

helmholtz-berlin.de

Authors

Daniel W. Paley −Columbia Nano Initiative, Columbia

University, New York 10027, United States

Susan Schorr −Department Structure and Dynamics of

Energy Materials, Helmholtz-Zentrum Berlin fur Materialien

und Energie GmbH, 14109 Berlin, Germany; Institute of

Geological Sciences, Freie Universität Berlin, 12249 Berlin,

Germany

Complete contact information is available at:

https://pubs.acs.org/10.1021/acs.jpclett.0c03722

Notes

The authors declare no competing ﬁnancial interest.

■ACKNOWLEDGMENTS

We acknowledge Dr. Trevor D. Hull and Professor Jonathan S.

Owen (Department of Chemistry, Columbia University,

United States) for providing single crystals of methyl-

ammonium lead bromide. X-ray diﬀraction was performed in

the Shared Materials Characterization Laboratory at Columbia

University. We thank Dr. Václav Petrícek (Fyzikální ustav AV

CR, Prague, Czech Republic) for providing us with a

development version of the program JANA2020 and

continuous support.

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