Dynamic nanodomains dictate macroscopic properties in lead halide perovskites

Synthesis of perovskite single crystals

MAPbBr₃: a mixture of 1-M PbBr₂ and 1-M MABr was dissolved in 1-ml DMF. To ensure complete dissolution, the solution was stirred vigorously at 25 °C for 6 h and then filtered with a 0.45-μm filter head before use. After adding a small MAPbBr₃ crystal to the filtered solution, the solution was transferred to an oven. To make the crystals larger, the solution was further heated to 85 °C at a rate of 10 °C per 30 min, and the crystal reached its full size after 24 h. The obtained crystals were separated and dried to obtain MAPbBr₃.

FAPbBr₃: a mixture of 1-M PbBr₂ and 1-M FABr was dissolved in 1-ml DMF/GBL (1:1). To ensure complete dissolution, the solution was stirred vigorously at 25 °C for 6 h and then filtered with a 0.45-μm filter head before use. After adding a small FAPbBr₃ crystal to the filtered solution, the solution was transferred to an oven. To make the crystals larger, the solution was further heated to 55 °C at a rate of 10 °C per 30 min, and the crystal reached its full size after 24 h. The obtained crystals were separated and dried to obtain FAPbBr₃.

FAPbBr₃ grown at Colorado State University: CH(NH₂)₂CH₃COO and HBr were obtained from Sigma-Aldrich. PbBr₂ and other solvents were procured from VWR and used without further purification. In a typical preparation, approximately 0.4 g of CH(NH₂)₂CH₃COO was dissolved in 8 ml of hydrobromic acid (47% v/v) at 80 °C for 15 min. Subsequently, 1.1 g of PbBr₂ (1.25:1.0 mole ratio of CH(NH₂)₂CH₃COO:PbBr₂) was added, and the solution was stirred until all the powder had dissolved. CH(NH₂)₂PbBr₃ was precipitated using ethanol as an antisolvent, and the powder was washed with ethanol. Single crystals were grown using an antisolvent method. A small vial containing 0.5 ml of a filtered 1-M solution of CH(NH₂)₂PbBr₃ in a 1:1 mixture of dimethylformamide and γ-butyrolactone by volume was placed in a sealed, larger vial containing approximately 5 ml of ethanol. Crystal growth reactions were carried out over three days.

Single-crystal X-ray diffuse scattering

We have performed various single-crystal diffraction experiments, using the following beamlines: MX1 (Australian Synchrotron), I19-1 (Diamond Light Source) and P21.1 (Deutsches Elektronen-Synchrotron); molybdenum (UNSW) and copper (Oxford) lab X-ray sources were also used. A brief summary of the methods is given below; Supplementary Note 2 provides the complete details.

Single-crystal perovskite samples were selected under a polarizing microscope (Leica M165Z) and picked up on a MicroMount (MiTeGen) consisting of a thin polymer tip with a wicking aperture. X-ray diffuse scattering measurements on MAPbBr₃ and FAPbBr₃ at 300 K and 200 K were carried out on the MX1 beamline at the Australian Synchrotron using X-rays of 12.9 keV with an X-ray flux of 36 × 10¹¹ s⁻¹ incident on an area of 120 μm × 120 μm. X-ray diffuse scattering measurements on MAPbBr₃ and FAPbBr₃ at 300 K and 200 K were carried out at the I19-1 beamline at the Diamond Light Source using X-rays of 18 keV with an X-ray flux of 1.542 × 10¹³ s⁻¹ incident on an area of 100 μm × 100 μm. X-ray diffuse scattering measurements on MAPbBr₃ and FAPbBr₃ were conducted at the P21.1 beamline at the Positron-Elektron-Tandem-Ring-Anlage (PETRA III) facility, Deutsches Elektronen-Synchrotron (the data from these measurements is presented in Fig. 1a,b). An X-ray beam with an energy of 101.45 keV (λ = 0.1222 Å) and a size of 0.35 × 0.35 mm² was used, delivering a flux of 2.5 × 10¹⁰ photons s^–1. Crystals were prepared with dimensions of approximately 500 × 500 × 500 μm³. X-ray diffuse scattering measurements on FAPbBr₃ at 300 K were carried out on a Rigaku Synergy S diffractometer fitted with a Dectris EIGER2 R 1M detector under copper radiation at the University of Oxford. Single-crystal diffraction measurements on MAPbBr₃ at 200 K and MAPbI₃ at 100 K were carried out on a Bruker D8 Quest single-crystal diffractometer with a PHOTON III detector using an IμS Incoatec Microfocus source with Mo Kα radiation (λ = 0.710723 Å) at UNSW.

The single crystals, mounted on the goniometer using a cryo-loop for intensity measurements, were coated with immersion-oil-type NVH and then quickly transferred to a nitrogen stream generated by an Oxford Cryostream 800 series. CrysAlisPro³⁹ was used for indexing, determination and refinement of the orientation matrix. In the process of data analysis, precession images were first unwrapped using CrysAlisPro. Detailed examination of the diffraction patterns revealed that the diffuse scattering adhered to Laue symmetry. Accordingly, Laue symmetry averaging was then applied to the data, also using CrysAlisPro.

MD simulations

To perform large-scale MD simulations of MAPbBr₃ and FAPbBr₃, Allegro^40,41 machine learning force fields were trained. The training, validation and test sets were constructed based on DFT using an on-the-fly structure selection process implemented in the Vienna ab initio simulation package^42,43, where a Gaussian-approximation-potential-style potential is fit on the fly, and training structures are picked from the MD simulations based on Bayesian error prediction⁴⁴. Structure selection runs were performed using the NPT ensemble in MD for each of the materials using 2 × 2 × 2 pseudo-cubic supercells at six separate temperatures, namely, 100 K, 160 K, 210 K, 270 K, 350 K and 450 K. The r²SCAN exchange–correlation functional⁴⁵, a plane-wave basis set with a cut-off energy of 500 eV and a 2 × 2 × 2 Γ-centred k-point grid were adopted. The energy threshold for electronic convergence was set to 10⁻⁵ eV, and a Gaussian smearing with a width of 50 meV was applied for the smearing of the electronic band occupancy. To avoid issues related to the incomplete basis set when large volume changes occur during the on-the-fly MD runs, an additional single-point DFT calculation was performed on all the structures, and these recalculated forces, energies and stresses made up the final training, validation and test sets. The full sets consisted of 2,985 and 2,640 structures for MAPbBr₃ and FAPbBr₃, respectively.

Separate Allegro machine learning force fields were trained for MAPbBr₃ and FAPbBr₃, using radial cut-offs of 6.5 Å, 2 layers and 32 tensor features with full O(3) symmetry and l_max = 2, a two-body latent multilayer perceptron with dimensions of [64, 128, 256, 512] and later latent multilayer perceptron with a dimension of [512], both with Sigmoid Linear Unit (SiLU) nonlinearities and a single-layer final edge-energy multilayer perceptron with a dimension of 128 and no linearity. Atomic distances were embedded using trainable Bessel functions. The models used the efficient mixed precision scheme described in ref. ⁴¹.

The training and validation sets contained 2,388 and 299 structures for MAPbBr₃ and 2,112 and 264 structures for FAPbBr₃, respectively, and were shuffled after each epoch. The training was performed using the Adam optimizer in PyTorch⁴⁶ for 1,858 and 2,141 epochs for MAPbBr₃ and FAPbBr₃, respectively, using a batch size of 5 and a learning rate of 0.001. A loss function with a 1:1:1 weighing of the Allegro per atom mean squared energy, force and stress terms, respectively, was used. The trained models achieved root mean squared errors on energies, force components and stress tensor components of 0.2 meV per atom, 10 meV Å^–1 and 0.5 kbar on hold out test sets containing 298 structures for MAPbBr₃ and corresponding values of 0.2 meV per atom, 7 meV Å^–1 and 0.3 kbar on hold out test sets containing 264 structures for FAPbBr₃. Parity and error distribution plots are provided in Supplementary Note 9.

The Allegro MD simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package⁴⁷, with the pair_allegro patch⁴⁸. Large simulation cells constructed as 20 × 20 × 20 pseudo-cubic unit cells were used for both materials and a 0.5-fs time step was used to integrate the classical equations of motion. For each material, two initial configurations were constructed, one with randomly oriented FA/MA molecules and one with perfectly aligned molecules. These initial configurations were equilibrated using fixed-shape NPT dynamics at 400 K for 200 ps, except for FAPbBr₃ with aligned molecules where 150-ps equilibration time was used. Then, for a specific temperature of interest, a further 50 ps of equilibration was performed before running 0.5 ns of NVE dynamics, resulting in 1 ns of production NVE trajectory data for each material at each temperature. Results were cross-checked between the runs with different initial configurations and no qualitative changes were observed, indicating that the structures had been sufficiently equilibrated with respect to the molecular orientations. On the basis of these MD trajectories, real space structural dynamics analysis was performed with the PDynA package²⁵.

To connect to the single-crystal X-ray diffuse scattering measurements, the dynamical structure factor S(q, E) was calculated from the MD trajectories using the pynamic structure factor package⁴⁹. For each material and initial configuration, the 0.5-ns trajectories were divided into 10 blocks of 50 ps. S(q, E) was then averaged over these blocks and (in fractional reciprocal lattice units) over the two initial configurations. We extracted S(q, E) for 0 ≤ H, K, L ≤ 5 and expanded these values to the whole plane by mirroring in the coordinate axes. Also, q-dependent atomic form factors, approximated as a sum of Gaussians, were used as described elsewhere²¹ and implemented in the pynamic structure factor package⁴⁹.

Inelastic neutron scattering

Constant-energy, variable-q (momentum) scans were performed using the thermal triple-axis spectrometer Taipan at the Australian Nuclear Science and Technology Organisation (ANSTO). Taipan was aligned with o–40′–40′–o collimation, in a configuration in which the incident neutron energy was varied with a fixed final scattered neutron energy of 14.87 meV. A graphite filter was used on the scattered side to remove higher-order scattering. The two large single crystals (about 1 cm³ in volume), MAPbBr₃ and FAPbBr₃ were aligned in the [H, H, L] and [H, K, 0] planes, respectively.

The cold triple-axis spectrometer SIKA at the ANSTO was aligned with o–60′–60′–60′ collimation using a large double-focusing pyrolytic graphite monochromator and analyser, which were oriented with a fixed final energy of 5 meV. This allowed an energy resolution of approximately 0.195 meV FWHM. The cooled Be filter was used to remove unwanted higher-order reflections. Samples were wrapped in a thin Teflon tape to prevent perovskite crystal surface from reacting with Al, and then mounted on an Al plate before being inserted into the cryostat. Counting times were approximately 5 min per point at the incident flux density of 1.5 × 10⁶ neutrons cm⁻² s⁻¹.

Hyperspectral PL microscopy

Wide-field, hyperspectral PL measurements were conducted using a Photon etc. IMA system. A ZEISS Plan-Neofluar objective lens with a numerical aperture (NA) of 0.75 and ×63 magnification was used for all the measurements. To reduce degradation caused by oxygen and humidity, samples were stored in a nitrogen-filled glovebox until just before measurement. For the temperature-dependent experiments, the samples were fixed with silver paste to the cold finger of an Oxford HiRes Microstat cooled with liquid helium. The sample was held at the set temperature for at least 15 min before every measurement. The samples were cooled down/heated up with a maximum rate of 1° min^–1 to avoid any unexpected thermal stress effects. A reference sample was used to calibrate the apparatus and determine the post-processing parameters necessary for correcting the image distortion caused by the optical elements in the detection path. Chromatic aberrations were mitigated by automatically changing the z position (focus) of the sample for every collected central wavelength based on a previously performed calibration measurement.

A 400-nm picosecond pulsed laser operating at a repetition rate of 100 kHz and a fluence of 0.68 μJ cm⁻² with a top-hat profile (size, 150 μm × 150 μm) was used as the excitation source for temperature-dependent PL measurements. This repetition rate was selected as the highest value at which no laser-induced decrease in PL intensity was observed in a vacuum, a common indicator of laser-induced damage. To ensure clean and consistent surface responses, single crystals were cleaved before each measurement. Using a microscope, PL was collected from a field of view of 85 μm × 85 μm, which was confirmed to be free of observable surface heterogeneities. Although hyperspectral microscopy enables the collection of PL spectra at every pixel, to improve the signal-to-noise ratio, particularly crucial at higher temperatures at which PL is weaker, the PL response was integrated across the entire field of view. This overall approach ensured that the measured PL represented the intrinsic properties of the material, minimizing contributions from surface morphology and contamination. This methodology is essential for accurately probing the material’s true optoelectronic response, as PL near rough surfaces or cracks can be substantially altered. Unlike macroscopic PL, which is usually collected from a larger area and lacks precise surface quality control, this approach provides a more reliable characterization of the intrinsic PL properties of single crystals.

The excitation laser was separated from the PL signal using a high-quality 405-nm Semrock dichroic mirror, effectively filtering the excitation light. The emitted light from the sample was then directed onto a volume Bragg grating, which dispersed it spectrally before being detected by a Hamamatsu ORCA-Flash4.0 V3 scientific complementary metal-oxide-semiconductor (sCMOS) camera. This camera features a 2,048 × 2,048 pixel² array, with each individual pixel measuring 6.5 μm × 6.5 μm.

For each objective lens, a two-step calibration process was conducted to determine the absolute number of photons at each point. Initially, a calibrated white light lamp was directed through the objective lens into an integrating sphere. By comparing the measured lamp spectrum at each point with its known spectrum, the system’s relative sensitivity was established both spectrally and spatially. In the second step, a 657-nm laser with a known optical power was fibre-coupled and imaged through the microscope. A hyperspectral measurement of the fibre output allowed for the conversion between detected counts and photons at this specific wavelength. By combining this absolute calibration with the relative calibration obtained from the white light lamp and integrating sphere, we accurately quantified the number of photons emitted across the spectrum at each point on the sample. More precisely, we determined the number of photons eV^–1 s^–1 cm^–2 sr^–1 detected per pixel (sr, steradians).

The details about PL FWHM, Urbach energy (E_U), quasi-Fermi level splitting (Δμ) (Supplementary Fig. 52) and the external PLQE measurements are provided in Supplementary Note 17.

Confocal PL diffusion measurements

Time-resolved PL diffusion measurements were performed using a confocal microscope setup (PicoQuant, MicroTime 200). Freshly cleaved crystals were excited by a 405-nm pulsed laser (PDL 828, PicoQuant, 10 MHz, 19–22 μJ per pulse, 11.6–13.4 μJ cm^–2, ~100-ps pulse width) focused through an air objective (×100, 0.9 NA). The emission signal was collected by the same objective and separated from the excitation light using a 405-nm longpass dichroic mirror (Z405RDC, Chroma).

To obtain the diffusion maps, the emission signals were scanned pixel by pixel using a Galvano scanner as the excitation source remained at a fixed position. Time-resolved PL signals subsequently passed through a 450-nm longpass filter and a 50-μm pinhole before being focused onto a PMA Hybrid 42 detector for time-correlated single-photon counting (time resolution, 100 ps).

The diffusion coefficients were extracted by tracking the expansion of the spatial PL distribution as a function of time, where we fit a Gaussian function to the PL profile at different time points. Since the observed diffusion behaviour exhibits a linear relationship between the squared standard deviation (σ²) and time (t), we determined the diffusion coefficient (D) according to the linear mean squared displacement model, where

On the basis of this model, we found the diffusion coefficients to be 0.40 cm² s^–1 and 0.27 cm² s^–1 for FAPbBr₃ and MAPbBr₃, respectively. The raw data and the corresponding fits are provided in Supplementary Fig. 47.

PDS

PDS measurements were conducted at room temperature using single crystal samples mounted inside a quartz cuvette filled with a thermo-optic liquid (3M Fluorinert FC-72). The excitation source was a halogen lamp paired with a 250-mm-focal-length grating monochromator, delivering tunable light-beam wavelengths for spectral scans. The light beam was modulated at 10 Hz using a mechanical chopper. The samples were excited using this monochromatic pump beam, generating heat through non-radiative recombination that induced an alternating temperature gradient at the sample surface. The PDS experiments were performed in the transverse configuration, with a continuous-wave probe laser beam (670 nm) passing parallel to the excitation area near the surface. The absorption-dependent deflection of this probe was detected using a quadrant silicon photodiode and measured synchronously using a lock-in amplifier (Stanford Research Systems SR830). This method provided a signal proportional to the absorbance, offering a high dynamic range and minimizing the scattering effects typically encountered in ultraviolet–visible spectroscopy. The E_U values were extracted from the absorption edge, at energies at which the absorption (A) is exponentially related to the photon energy according to

where A₀ is the optical absorption coefficient and E_g is the bandgap energy. For each crystal, the measurements were performed at three different spots to ensure accuracy.

Low-temperature optical microscopy

Samples were measured in an Oxford HiRes Microstat cooled using liquid helium and kept in a vacuum during the measurement. The emergence of ferroelastic nanodomains was observed using the Photon Etc. IMA microscopy system. Here ×20 (Nikon TU Plan Fluor, 0.45 NA) and ×100 long-working-distance objective (Nikon TU Plan Fluor, 0.8 NA) with appropriate chromatic aberration corrections were used for all measurements. Single crystals were cleaved immediately before measurements to obtain a fresh and uniform surface and then mounted in an open-cycle liquid-helium cryostat. The samples were cooled down/heated up with a maximum rate of 1° min^–1 to avoid any unexpected thermal stress effects. The images were taken using a high-sensitivity camera (Hamamatsu ORCA-Flash4.0 V3 sCMOS camera) with 2,048 × 2,048 pixels² (each pixel, 6.5 × 6.5 μm²) that was thermoelectrically cooled to –10 °C.

Differential scanning calorimetry

Differential scanning calorimetry measurements were performed on a Netzsch DSC Proteus 204 F1 device in an Al crucible under a N₂ atmosphere. The samples were cooled and heated in the temperature range of 115–330 K with a scan rate of 10 K min^–1. The masses of the MAPbBr₃ and FAPbBr₃ samples were 19.9 mg and 20.3 mg, respectively.