Collective Diffraction Effects in Perovskite Nanocrystal Superlattices

Abstract

ConspectusFor almost a decade now, lead halide perovskite nanocrystals have been the subject of a steadily growing number of publications, most of them regarding CsPbBr₃ nanocubes. Many of these works report X-ray diffraction patterns where the first Bragg peak has an unusual shape, as if it was composed of two or more overlapping peaks. However, these peaks are too narrow to stem from a nanoparticle, and the perovskite crystal structure does not account for their formation. What is the origin of such an unusual profile, and why has it been overlooked so far? Our attempts to answer these questions led us to revisit an intriguing collective diffraction phenomenon, known for multilayer epitaxial thin films but not reported for colloidal nanocrystals before. By analogy, we call it the multilayer diffraction effect.Multilayer diffraction can be observed when a diffraction experiment is performed on nanocrystals packed with a periodic arrangement. Owing to the periodicity of the packing, the X-rays scattered by each particle interfere with those diffracted by its neighbors, creating fringes of constructive interference. Since the interfering radiation comes from nanoparticles, fringes are visible only where the particles themselves produce a signal in their diffraction pattern: for nanocrystals, this means at their Bragg peaks. Being a collective interference phenomenon, multilayer diffraction is strongly affected by the degree of order in the nanocrystal aggregate. For it to be observed, the majority of nanocrystals within the sample must abide to the stacking periodicity with minimal misplacements, a condition that is typically satisfied in self-assembled nanocrystal superlattices or stacks of colloidal nanoplatelets.A qualitative understanding of multilayer diffraction might explain why the first Bragg peak of CsPbBr₃ nanocubes sometimes appears split, but leaves many other questions unanswered. For example, why is the split observed only at the first Bragg peak but not at the second? Why is it observed routinely in a variety of CsPbBr₃ nanocrystals samples and not just in highly ordered superlattices? How does the morphology of particles (i.e., nanocrystals vs nanoplatelets) affect the appearance of multilayer diffraction effects? Finally, why is multilayer diffraction not observed in other popular nanocrystals such as Au and CdSe, despite the extensive investigations of their superlattices?Answering these questions requires a deeper understanding of multilayer diffraction. In what follows, we summarize our progress in rationalizing the origin of this phenomenon, at first through empirical observation and then by adapting the diffraction theory developed in the past for multilayer thin films, until we achieved a quantitative fitting of experimental diffraction patterns over extended angular ranges. By introducing the reader to the key advancements in our research, we provide answers to the questions above, we discuss what information can be extracted from patterns exhibiting collective interference effects, and we show how multilayer diffraction can provide insights into colloidal nanomaterials where other techniques struggle. Finally, with the help of literature patterns showing multilayer diffraction and simulations performed by us, we demonstrate that this collective diffraction effect is within reach for many appealing nanomaterials other than halide perovskites.

Figure 1

XRD patterns of different CsPbBr₃ samples. (a) XRD patterns of CsPbBr₃ superlattices (green), nanocrystal powders (red), and bulk powders (black, orthorhombic Pnma reference in gray). Peaks from nanocrystal powders retain the positions and relative intensities of the bulk but are broadened due to the nanometric size. In the superlattices pattern most peaks are suppressed due to preferred orientation, and the first Bragg peak (yellow shaded area) is visibly split. (b) Close-up of the first superlattice peak plotted on the q scale (top). Superlattice fringes are sometimes confused with the cubic (100) → orthorhombic (020)/(101)/(101̅) CsPbBr₃ peak split, shown on the bulk pattern for comparison (bottom). (c) Optical microscopy image of CsPbBr₃ nanocube superlattices. Adapted with permission from refs (1) and (2). Copyright (2019, 2021) American Chemical Society.

Figure 2

Principles of multilayer diffraction. A nanocrystal superlattice (a) is composed of (b) alternating inorganic (blue) and organic layers (black), both contributing to the superlattice periodicity Λ (red). Each layer has a scattering factor for nanocrystals (F_NC) and for organic ligands (F_L), computed via eqs 1 and 2 respectively. A multilayer diffraction pattern (c, red line) stems from the constructive interference of many such organic–inorganic bilayers. Fringes are affected by the superlattice periodicity (position, q_n = 2πn/Λ), superlattice disorder σ_L (width), and scattering factors (intensity). F_NC is predominant due to the high electron density in nanocrystals, resulting in the intensity of fringes being modulated by the nanocrystal Bragg peaks (c, black dashed line). Similarly, a stack of nanoplatelets (d) is a multilayer (e) of alternating inorganic (cyan) and organic layers (black), resulting in the nanoplatelet scattering factor F_NP modulating the intensity of the multilayer diffraction fringes (f). The only difference between (a–c) and (d–f) is that F_NC is computed based on unit cells, while for F_NP one must consider individually each atomic layer within the platelet (f₀... f_J–1). Adapted with permission from refs (2) and (3). Copyright (2021) American Chemical Society.

Figure 3

Multilayer diffraction for nanoparticle characterization. (a) Comparing patterns of samples prepared with different ligands can give insight into the surface passivation of nanocrystals. Replacing the carboxylic acid in the synthesis of perovskite nanoplatelets leaves their pattern unchanged, while different amines result in different stacking periodicities, demonstrating that platelets are passivated by amines only. (b) Linear regression allows the extraction of Δq and therefore Λ. (c) Evolution of the first Bragg peak of CsPbBr₃ nanocube superlattices upon thermal annealing (d). (e–g) Evolution of nanocrystal thickness, interparticle distance, and stacking disorder σ_L tracked by multilayer diffraction. (h, i) The surface termination of perovskite nanoplatelets is identified by comparing their diffraction pattern with simulations. CsBr and PbBr₂ terminations (i) are excluded due to mismatching fringe intensities, while an oleylammonium bromide termination (h) matches the pattern. (h) A quantitative profile fit allows refining the vertical atomic coordinates and the surface coverage factor. Adapted with permission from refs (2) and (3). Copyright 2021 American Chemical Society.

Figure 4

Simulated multilayer diffraction patterns. (a) Cs₄PbBr₆ nanospheres oriented with (012) planes parallel to the substrate and packed in different geometries. From top to bottom: simple cubic, BCC, FCC, and HCP. (b) Sphalerite CdSe nanoplatelets of different thicknesses, indicated as Cd–Se monolayers (ML). The diffraction profile of a single nanoplatelet (= |F_NP|²) is traced by a black dashed curve. (c) Simple cubic superlattices of CsPbBr₃ and PbS nanocubes and their mixtures. (d) Stacks of CsPbBr₃ and PbS nanoplatelets and their mixtures. If two kinds of nanocrystals do not mix and form segregated superlattices, the the resulting pattern will simply be the sum of patterns for single-material superlattices. All patterns are plotted on the 2θ_Cu Kα scale and include instrumental intensity corrections (thin-film Lorentz polarization, instrumental broadening). See SI, section S4 for details.

Figure 5

Literature patterns showing multilayer diffraction. Colors identify classes of materials. (a) Yellow: metal-halide perovskites. (b) Green: synthetic 2D materials. (c) Red: metal–organic precursors for colloidal syntheses. (d) Blue: metal oxides and hydroxides. Numbers within black labels indicate the corresponding reference. All patterns were digitized with WebPlotDigitizer and are plotted on the 2θ_Cu Kα scale. Abbreviations: OA = oleylamine; RP = Ruddlesden–Popper; T = (F, OH, O); BU = butyl-; SULF = sulfate. Data are taken from refs (24), (33), (34), (50), (52), (−57), and (−64).