2021

[1,2]\fnmNicholas J \surWeadock \equalcontThese authors contributed equally to this work. \equalcontThese authors contributed equally to this work.

[3]\fnmDmitry \surReznik

[1,2,14]\fnmMichael F \surToney

1]\orgdivMaterials Science and Engineering, \orgnameUniversity of Colorado, Boulder, \orgaddress\cityBoulder, \stateCO, \postcode80309, \countryUSA

2]\orgdivDepartment of Chemical and Biological Engineering, \orgnameUniversity of Colorado, Boulder, \orgaddress\cityBoulder, \stateCO, \postcode80309, \countryUSA

3]\orgdivDepartment of Physics, \orgnameUniversity of Colorado, Boulder, \orgaddress\cityBoulder, \stateCO, \postcode80309, \countryUSA

4]\orgdivDepartment of Chemical Engineering, \orgnameStanford University, \orgaddress\cityStanford, \stateCA, \postcode94305, \countryUSA

5]\orgdivDepartment of Chemistry, \orgnameStanford University, \orgaddress\cityStanford, \stateCA, \postcode94305, \countryUSA

6]\orgdivDepartment of Mechanical Science & Engineering and Materials Resesarch Laboratory, \orgnameUniversity of Illinois at Urbana-Champaign, \orgaddress\cityUrbana, \stateIL, \postcode61801, \countryUSA

7]\orgdivAdvanced Photon Source, \orgnameArgonne National Lab, \orgaddress\cityLemont, \stateIL, \postcode60439, \countryUSA

8]\orgdivNeutron Scattering Division, \orgnameOak Ridge National Laboratory, \orgaddress\cityOak Ridge, \stateTN, \postcode37830, \countryUSA

9]\orgdivISIS Facility, \orgnameRutherford Appleton Laboratory, \orgaddress\cityChilton, \stateDidcot, \postcodeOxon OX11 0QX, \countryUnited Kingdom

10]\orgdivDepartment of Physics, \orgnameRoyal Holloway University of London, \orgaddress\postcodeEgham TW20 0EX, \countryUnited Kingdom

11]\orgdivNIST Center for Neutron Research, \orgnameNational Institute of Standards and Technology, \orgaddress\cityGaithersburg, \stateMD, \postcode20899, \countryUSA

12]\orgdivDepartment Chemie, \orgnameUniverstät Paderborn, \orgaddress\cityPaderborn, \countryGermany

13]\orgdivStanford Institute for Materials and Energy Sciences, \orgnameSLAC National Accelerator Laboratory, \orgaddress\cityMenlo Park, \stateCA, \postcode94025, \countryUSA

14]\orgdivRenewable and Sustainable Energy Institute (RASEI), \orgnameUniversity of Colorado, Boulder, \orgaddress\cityBoulder, \stateCO, \postcode80303, \countryUSA

The crystal structure and associated symmetry of a material are key determinants of mechanical, electronic, optical, and thermal properties. One has to look no further than seminal condensed matter physics textbooks for derivations of these properties made possible by consideration of the long-range or average order determined by the crystal lattice and translational symmetry ashcroftSolidStatePhysics2000 ; kittelIntroductionSolidState2019 . However, in many materials including disordered rocksalts and intercalation compounds used for battery cathodes, relaxor ferroelectrics, thermoelectrics, and oxide and halide perovskites, the important properties are not well described by the long-range structure. Instead it is short-range order that dominates aspects of the structure-function relationship clementCationdisorderedRocksaltTransition2020 ; krogstadReciprocalSpaceImaging2020a ; simonovHiddenDiversityVacancy2020a ; xuThreedimensionalMappingDiffuse2004 ; krogstadRelationLocalOrder2018 ; rothSimpleModelVacancy2020a ; lanigan-atkinsTwodimensionalOverdampedFluctuations2021 . In scattering experiments, short-range or local order manifests as diffuse scattering; a result of static and dynamic deviations from the average structure. Fixed chemical or local structural correlations result in static diffuse scattering, whereas thermal diffuse scattering arises from dynamic displacements due to lattice dynamics welberryOneHundredYears2016 . Resolving structural correlations in disordered materials has recently become more feasible with the development of high flux, single crystal X-ray and neutron diffuse scattering instruments and sophisticated modeling algorithms rosenkranzCorelliEfficientSingle2008 ; yeImplementationCrossCorrelation2018 ; krogstadReciprocalSpaceImaging2020a ; proffenAdvancesTotalScattering2009 ; simonovYellComputerProgram2014 ; morganRmcdiscordReverseMonte2021 , opening up enormous opportunities to understand how local order impacts materials properties.

Organic-inorganic metal halide perovskites are a recently re-invigorated class of semiconductors with remarkable optoelectronic performance that defies traditional intuition: Lead-based metal halide perovskites (LHPs) possess a mechanically soft, defect-tolerant crystal lattice with strong structural disorder and mobile ions at modest temperatures eggerWhatRemainsUnexplained2018 . Fluctuations in the orbital overlaps arising from large thermal displacements of iodide in methylammonium (CH₃NH${}_{3}^{+}$, CD₃ND${}_{3}^{+}$ = MA) lead iodide directly influence the temperature dependence of the electron (or hole) mobility and optical bandgap mayersHowLatticeCharge2018a . X-ray and neutron diffraction measurements, which probe long-range order, report that the high-temperature phase is cubic with well-defined Bragg peaks wellerCompleteStructureCation2015 ; whitfieldStructuresPhaseTransitions2016 ; swainsonPhaseTransitionsPerovskite2003 . This cubic perovskite structure is shown in Figure 1a and consists of corner-sharing PbX₆ (X = I^-, Br^-) octahedra surrounding a dynamically disordered MA⁺ cation within the cuboctahedral interstice. Measurements probing short-range order in the high-temperature phase, however, suggest the local structure is of lower symmetry beecherDirectObservationDynamic2016 ; cominLatticeDynamicsNature2016 ; lauritaChemicalTuningDynamic2017 ; zhaoPolymorphousNatureCubic2020 ; weadockTestDynamicDomainCritical2020 . The lack of consensus regarding the exact symmetry and dynamics of this enigmatic local structure limits our understanding and control of optoelectronic properties and ion migration in LHPs. Short-range order arising from dynamic two-dimensional correlations of lead bromide octahedra has recently been identified in CsPbBr₃ lanigan-atkinsTwodimensionalOverdampedFluctuations2021 . These correlations are most prominent in the cubic phase above 433 K and may not play a significant role in CsPbBr₃-based device operation. The significance and prevalence of such correlations in hybrid LHPs, including contributions from the organic cations, is not resolved.

Ion migration in LHPs, especially under illumination, is detrimental to device performance and stability yet the origin is not well understood. Consequences of ion migration include formation of space charge potentials at interfaces, accelerated degradation and loss of constituent elements, and light-induced phase segregation in mixed-halide compositions yuanIonMigrationOrganometal2016 . A complete picture of the high-temperature structure and dynamics is essential to model ion migration pathways accurately holekevichandrappaCorrelatedOctahedralRotation2021 . It is also important to characterize the short-range order on the organic sublattice, as the rotating MA⁺ molecular dipoles may screen band-edge charge carriers and extend carrier lifetimes chenOriginLongLifetime2017 .

We use single crystal X-ray and neutron diffuse scattering and neutron spectroscopy, combined with molecular dynamics (MD) simulations, to uncover the true structure and dynamics of the nominally simple cubic ($Pm\bar{3}m$) phases of MAPbI₃ ($>$¿ 327 K) and MAPbBr₃ ($>$¿ 237 K). We find that the cubic phase (Fig. 1a, c) is comprised of dynamic, two-dimensional roughly circular “pancakes” of tilted lead halide octahedra which align along any of the three principal axes of the cubic structure and (Fig.  1b, d) induce additional structural correlations of the organic sublattice in the layers sandwiching the octahedra. Taken together, these regions of dynamic local order are several unit cells in diameter with lifetimes on the order of several picoseconds, resulting in a dynamic landscape for charge carriers and ion migration. This extended dynamic local order was neither observed nor predicted in previous studies of structural dynamics in hybrid LHPs ferreira_elastic_2018 ; gold-parkerAcousticPhononLifetimes2018 ; leguyDynamicsMethylammoniumIons2015 ; chenRotationalDynamicsOrganic2015 . Embedded within this dynamic structure we find additional static 3D droplets of the intermediate tetragonal phase, consistent with previous reports weadockTestDynamicDomainCritical2020 ; cominLatticeDynamicsNature2016 . The structural correlations on the MA⁺ sublattice are a unique characteristic in these hybrid systems with implications discussed below.

The experimental scattering function $S(\mathbf{Q})$ from the cubic phase of MAPbI₃ at 345 K is shown in Fig. 2. Specifically, we present the L = 0.5, 1.5 planes measured with both X-ray (XDS, Fig. 2a,c, left panels) and neutron diffuse scattering (NDS, Fig 2b,d, left panels). $S(\mathbf{Q})$ calculated from atomic trajectories obtained with MD simulations of MAPbI₃, see Methods, is plotted in the right-hand panels of a-d and both panels of g,h for comparison.

The experimental $S(\mathbf{Q})$ of the cubic phase of MAPbBr₃ at 250 K is plotted in Fig. S1 and show diffuse scattering intensity profiles for XDS and NDS that are nearly identical to MAPbI₃.

Our approach to solving the true structure of the nominally simple cubic MAPbI₃ and MAPbBr₃ involves decomposing the observed diffuse scattering profile into four components and analyzing their energy and Q-dependence. These components include: (1) rods of constant (XDS) or varied (NDS) intensity spanning the BZ edge (M-R direction in Fig. 1, $q=[0.5,0.5,L]$); (2) an additional contribution centered at the R-point [$q=(0.5,0.5,0.5)$]; (3) broad intensity centered at the X-point [$q=(0.5,0,0)$] observed previously and discussed in the SI; and (4) contributions from the MA⁺ sublattice deduced from the alternating diffuse rod intensity profile observed in NDS but not XDS. This difference between XDS and NDS is highlighted with one-dimensional line cuts of XDS and NDS data shown in Figs. 2e and f. Specifically, the peaks at K = $\pm 2.5$ originating from the extended rod along H in XDS are not observed in the NDS data. Our analysis is complemented by MD simulations of MAPbI₃ which reproduce the experimental diffuse scattering. The experimental and calculated XDS (Fig. 2) show remarkable agreement. Comparing the two results in a root-mean-square error of 3.9% (see Figs. S2,3 and related discussion) after adjusting only the background and scaling of the calculated $S(\mathbf{Q})$. The agreement between the experimental and calculated NDS is very good, however there are small differences in intensities highlighted in Fig. S4 that likely arise from the classical potentials used here. Given the good agreement, we examine the real-space structure simulated with MD to determine the origin of the diffuse scattering.

The XDS rods of diffuse scattering have constant intensity along the entire BZ edge (Fig. S5) and a width larger than the instrument resolution ($\approx 0.02\AA^{-1}$). This lineshape arises from two-dimensional structural correlations in real-space. We show these correlations in Fig. 3, which presents a simplified visualization of the MD simulation. These plots track PbI₆ octahedral rotations, defined by azimuthal rotation angle $\phi$ about the [001] (Fig. 3a,b) or [010] (Fig. 3c,d) directions. Correlated regions of alternating large tilts are identified by neighboring red and blue pixels (representing octahedra) in Fig. 3a and d. Along the axis of rotation indicated in the corresponding schematic the tilts are uncorrelated between planes (Fig. 3b,c), revealing the existence of two-dimensional pancakes of tilted octahedra with a diameter on the order of 5 unit cells. These regions are reminiscent of tilts associated with the tetragonal structure in Fig. 1b. The tilts generate in-plane antiphase structural correlations with a periodicity of $2a$. The associated wave vector of the correlation is $q=2\pi/2a=\pi/a$, which is the zone boundary for the simple cubic BZ. The tilted regions are confined to a single PbI₆ sheet and therefore the diffuse scattering intensity manifests as rods.

The same structural correlations are present in the cubic phase of MAPbBr₃, since the XDS and NDS are qualitatively identical to that of MAPbI₃ (Fig. S1, 2). Extended diffuse rods observed in the all-inorganic CsPbBr₃ lanigan-atkinsTwodimensionalOverdampedFluctuations2021 point to the existence of two-dimensional structural correlations of lead halide octahedra in all LHPs with a high temperature cubic structure.

Intermolecular structural correlations in the MA⁺ sublattice are implied from the alternating intensity pattern of the diffuse rods present in NDS but not XDS. The MA⁺ correlations are explored in the calculated $S(\mathbf{Q})$ in Figs. 2g,h, and S6-8. Specifically, we set the neutron scattering lengths of Pb and I to zero in the calculation to isolate the contribution from the MA⁺ cation (Figs. 2h, S6c, g, k and S7), and C, N, and D to zero to isolate contributions from the inorganic octahedra (Figs. 2g, S6b, f, j and S7). Diffuse scattering from the inorganic framework manifests as extended diffuse rods along the zone edge, resembling the experimental XDS as the X-ray atomic form factors for C, N, D are small compared to those for Pb and I and contribute little intensity. The calculated $S(\mathbf{Q})$ from CD₃ND${}_{3}^{+}$ shows remarkable behavior (Figs. 2h, S6, S7), namely well-defined diffuse rods along the zone edges in addition to broad, isotropic intensity. This broad, isotropic component is a result of uncorrelated MA⁺ dynamics leguyDynamicsMethylammoniumIons2015 ; chenRotationalDynamicsOrganic2015 commonly associated with the nominally cubic phases of LHPs. The presence of additional zone edge intensity, however, shows that previously unreported intermolecular structural correlations exist on the MA⁺ sublattice. These local correlations are not purely 2D as the intensity varies along individual diffuse rods in the MA⁺ $S(\mathbf{Q})$ (Fig. S6,7). Layers of PbX₆ octahedra are sandwiched between layers of MA⁺ cations, therefore we expect the out-of-plane MA⁺ structural correlations to extend at least two unit cells. Finally, the MA⁺ and PbX₆ correlations are connected as evidenced by a nonzero interference term, shown in Fig. S8, which results in the alternating diffuse intensity profile observed in NDS.

We propose that MA⁺ orientation is driven by PbX₆ tilts; the MA⁺ molecules orient to favor the lowest energy configuration defined by the distorted cuboctahedral geometry of the tilted regions and electrostatic interactions between lead halide octahedra and polar MA⁺ molecules zhu2019mixed ; lahnsteiner2016room . The existence of 2D correlations on the PbBr₆ sublattice in CsPbBr₃ supports this prediction as the Cs⁺ cations are not expected to influence the PbBr₆ octahedra through hydrogen bonding. lanigan-atkinsTwodimensionalOverdampedFluctuations2021 .

The lateral size of 2D pancakes, given by the correlation length, $\xi$, is obtained from the $\mathbf{Q}$-linewidth of the XDS and NDS diffuse rods. We fit the intensity of one-dimensional cuts in $\mathrm{\AA}^{-1}$ across various diffuse rods to a Lorentzian lineshape in accordance with Ornstein-Zernike theory for exponentially decaying spatial correlations:

where $\Gamma=1/\xi$ is the half-width at half-maximum (HWHM) of the fitted Lorentzian and $\xi$ is the correlation length xuThreedimensionalMappingDiffuse2004 ; valeCriticalFluctuationsSpin2019 . Correlation lengths of 4-6 unit cells ($\sim$3 nm) in diameter are obtained for MAPbBr₃ and MAPbI₃ as reported in Table S1. Similar $\xi$ are obtained from NDS and XDS, indicating the in-plane $\xi$ of the 2D pancakes are nearly equivalent for the PbX₆ and MA⁺ sublattices. We also calculate $\xi$ directly from the MD simulations, see Figs. S9-12 and related discussion, and find values consistent with experiment.

Molecular dynamics simulations show that the 2D structural correlations are transient and diffusive, with dynamics of the octahedral correlations tracked in Fig. 3 and Supplementary Videos 1 and 2. We investigate the dynamics of the PbI₆ and MA⁺ correlations by evaluating the energy dependence obtained from the dynamical structure factor $S(\mathbf{Q},E=\hbar\omega)$ measured with neutron inelastic spectroscopy (INS) on MAPbI₃ at 340 K. The $S(\mathbf{Q},E)$ in the L = 0.5 plane, integrated from $-1\leq E\leq 1$ meV is shown in the right-hand panel of Fig. 4a and matches that observed with NDS (Fig. 2b) and that calculated from MD (left-hand panel of Fig. 4b). A slice of $S(\mathbf{Q},E)$ along the diffuse rod presented in Fig. 4b shows that the intensity is centered at $E=0$ meV with no inelastic component visible. Incoherent-background-subtracted energy scans (integrated along the length of the diffuse rod) show a finite, quasielastic component as exemplified in Fig. 4d. Quasielastic scattering is a result of diffusive or relaxational motions involving small energy transfers, manifesting as a peak centered at $E=0$ meV with a non-zero linewidth. The scattering is fit to a relaxational (Lorentzian) model, broadened by the resolution function, as shown in Fig. 4d. The average energy HWHM, plotted in Fig. 4e, is $0.20\pm 0.01$ meV with a corresponding lifetime $\tau=\hbar/\mathrm{HWHM}$ of $3.25\pm 0.13$ ps. Lifetimes are obtained in the same manner from the calculated $S(\mathbf{Q},E)$ as shown in Fig. S13, and the average lifetime of 6.3 ps is reported in Fig. 4e. We note that these lifetimes are consistent with the mean residence time for rotations of the C-N bond in cubic MAPbI₃, suggesting that MA⁺ reorientations are in part driven by the PbX₆ correlations chenRotationalDynamicsOrganic2015 . Overall, the structural correlations are dynamic with finite lifetime of several picoseconds.

R-point scattering ($\mathbf{Q}=[1.5,1.5,0.5]$, Fig. 4c) noted above has been investigated previously and is attributed to static droplets of the intermediate-temperature tetragonal phase embedded within the high-temperature cubic phase, possibly nucleating about defects weadockTestDynamicDomainCritical2020 . We observe no inelastic scattering at the R-points in both the INS or MD $S(\mathbf{Q}=(H,1.5,0.5),E)$ in Fig. 4b. Incoherent-background-subtracted energy scans at a constant Q corresponding to R-points are best fit by a Gaussian lineshape with resolution limited linewidth (Fig. 4c), hence this scattering is static with $\xi=2$ nm (Table S1), consistent with previous reports weadockTestDynamicDomainCritical2020 . These droplets are distinct from the 2D pancakes but we are not able to resolve the spatial distribution of these two components. We expect that both structures influence MA⁺ orientation.

The appearance of small regions of a lower temperature phase above the phase transition temperature is often a hallmark of critical scattering. For phase transitions with critical fluctuations, there is a marked jump in intensity of the associated low temperature phase as the system is cooled through the transition stirling_critical_1996 . We investigate the temperature dependence of the diffuse scattering in Figs. S13,14 through the cubic-tetragonal transition temperatures for MAPbI₃ and MAPbBr₃. The intensity of the diffuse rods increases continuously with decreasing temperature through the transition, however no significant and abrupt jump in intensity is observed. Furthermore, the two-dimensional nature of the dynamic structural correlations is not typical of critical behavior at phase transitions. For the R-point scattering, however, there is a large jump in intensity as the cubic R-point transforms to the $\Gamma$-point of the tetragonal phase, giving Bragg scattering. The critical nature of the R-point scattering has been addressed previously weadockTestDynamicDomainCritical2020 ; cominLatticeDynamicsNature2016 .

Our results show that the simple cubic phase of MAPbI₃ and MAPbBr₃ is in fact a composite structure containing dynamic 2D structural correlations and static tetragonal droplets with $\xi$ of a few nanometers. MA⁺ correlations, induced by octahedral tilts, extend 2-3 unit cells in the normal direction. The simple cubic phase is only recovered when this composite is averaged over space and time (Fig. 1 a-d). Structural fluctuations of lead halide octahedra generate large variations in the optical bandgap and electron-phonon coupling lanigan-atkinsTwodimensionalOverdampedFluctuations2021 ; mayersHowLatticeCharge2018a ; zhaoPolymorphousNatureCubic2020 , therefore it is essential to incorporate the observed correlations when modeling optoelectronic properties. We expect the dynamic correlations and static droplets to introduce spatial anisotropy to the electronic potential landscape with a longer lifetime than variations resulting from uncorrelated dynamic disorder. Furthermore, we must consider the effect of local correlations of the organic sublattice on LHP properties. MA⁺ has a large electric dipole moment of 2.3 Debye along the C-N axis chenOriginLongLifetime2017 . Uncorrelated dynamic disorder of the MA⁺ sublattice, as previously reported, would generate a zero or negligible dipole moment. The MA⁺ correlations observed here, however, may generate a net dipole moment resulting in transient, local ferroelectric or antiferroelectric domains with lifetimes on the order of 3-6 ps. These domains influence electron-hole recombination and shift band alignment biEnhancedPhotovoltaicProperties2017 ; liuChemicalNatureFerroelastic2018 . The dynamic structural correlations also affect ion migration. In solid-state ion conductors, rotational dynamics of polyanions are known to strongly impact cation mobility smithLowtemperaturePaddlewheelEffect2020 . Since the lifetime of the dynamic structural correlations (3-6 ps) exceeds the calculated time between attempted halide jumps (1 ps) frostWhatMovingHybrid2016 , a prefactor in calculating diffusivity, the correlations appear static and must be considered in future calculations. These correlations may impose a barrier to halide migration and reduce diffusivity holekevichandrappaCorrelatedOctahedralRotation2021 .

The two-dimensional dynamic structural correlations we observe may arise from the strongly anharmonic lattice dynamics observed in LHPs zhu2019mixed ; gold-parkerAcousticPhononLifetimes2018 ; songvilayCommonAcousticPhonon2019 ; ferreira_elastic_2018 . Indeed, the structural correlations in CsPbBr₃ are reportedly driven by soft, overdamped, anharmonic phonons at the BZ edge lanigan-atkinsTwodimensionalOverdampedFluctuations2021 . We do not observe dispersive phonons along the BZ edge M-R branch in Fig. 4b, therefore these modes may be overdamped for MAPbI₃ and MAPbBr₃ as well. The structural correlations we observe on the PbX₆ sublattice have a longer lifetime than those in CsPbBr₃, and are prevalent at device relevant temperatures. The additional intermolecular correlations between MA⁺ are not present in CsPbBr₃ and contribute to the improved optoelectronic properties of hybrid LHPs.