Electron ptychography reveals correlated lattice vibrations at atomic resolution

Article https://doi.org/10.1038/s41467-026-74135-4 Electron ptychography reveals correlated lattice vibrations at atomic resolution Received: 1 October 2025 Accepted: 29 May 2026 1234567890():,; 1234567890():,; Check for updates Anton Gladyshev 1 , Benedikt Haas1, Thomas C. Pekin Marcel Schloz1, Peter Rez 4 & Christoph T. Koch1 1 , Tara M. Boland2,3, Electron Ptychography is a computational imaging technique capable of performing phase retrieval at atomic resolution. Here we introduce the CAVIAR framework (Correlated Atomic Vibration Imaging with sub-Angstrom Resolution) that reveals spatial correlations in atomic displacements at the atomic scale. Using realistically simulated data for a symmetric Σ9 grain boundary in silicon and experimental data of a hexagonal boron nitride bicrystal, we observe correlations between atomic movements in the range of 10-20 pm at room temperature in agreement with our expectation. From only the atomic masses and temperature as input, we obtain frequencies of the longitudinal and transverse acoustic and optic phonons from just a few nm3 volume, in agreement with inelastic neutron scattering. This ability to spatially resolve correlated atomic motion distinguishes CAVIAR and positions it as a complementary tool to vibrational electron energy loss spectroscopy for exploring atom dynamics at the finest scale. While capable of imaging the lateral positions of atomic columns constituting thin slabs of material, the achievable resolution of conventional electron imaging techniques in a transmission electron microscope (TEM) is very sensitive to instrumental imperfections: the partial coherence of the electron source, lens aberrations as well as mechanical and electronic instabilities of the microscope. Ptychography1–5 is a computational phase retrieval technique that, to some extent, can compensate for the imperfections of the equipment. Numerous quite different variations of this technique exist, e.g., Fourier and near-field ptychography6,7, and a variety of reconstruction schemes for them. The method used in this paper, “classical” far-field ptychography, recovers a complex transmission function of a specimen from a four-dimensional scanning transmission electron microscopy (4D-STEM) dataset containing transmitted intensities8 collected with a pixelated detector while illuminating overlapping areas of its surface with a convergent beam. Ptychography has proven itself as a powerful tool for imaging of thin samples5,9. Recently, it was shown that adopting a multislice formalism makes it possible to resolve specimen features as fine as the blurring due to the vibrations of atoms10, an achievement that triggered a rapid rise in popularity of electron ptychography in the community. It was demonstrated that spatially non-uniform vibrational modes can be fitted to the atomic shapes produced via ptychography11. However, fully surpassing this limit of atomic vibrations was not possible, since a coherent specimen model as depicted in Fig. 1a cannot describe the formation of an incoherent diffraction pattern of the type schematically shown in Fig. 1c. Previously, there were attempts to include an incoherent sample model in experiments with laser12 and X-ray illumination13,14, but applications to electron ptychography assumed only uncorrelated atomic vibrations15,16. Here we propose to utilize thermal diffuse scattering (TDS)17 as a source of information about correlated atomic movement rather than the ultimate limiting factor in achievable spatial resolution10. This information can be retrieved using the ptychographic reconstruction scheme CAVIAR (Correlated Atomic Vibration Imaging with sub-Å ngstrom Resolution), presented here, which is based on mixed-object electron ptychography. It combines a mixed-object formalism14, where the specimen is modeled as a statistical ensemble of states, with lattice Green’s function analysis18, a technique well-established in molecular dynamics (MD) for quantifying vibrational correlations. The evolution 1 Department of Physics, Humboldt-Universität zu Berlin & Center for the Science of Materials Berlin, Berlin, Germany. 2School for Engineering of Matter Transport and Energy, Arizona State University, Tempe, AZ, USA. 3Computational Atomic-Scale Materials Design (CAMD), Technical University of Denmark, e-mail: Kgs. Lyngby, Denmark. 4Department of Physics, Arizona State University, Tempe, AZ, USA. Nature Communications | (2026)17:5173 1 Article https://doi.org/10.1038/s41467-026-74135-4 a b Source with Finite Width c Incoherent Superposition of e − Beams Sample Con guration 2 d Con guration 1 Con guration 1 2D Ptychography 3D Multislice Ptychography Incoherent Average over Available Con gurations 4D Mixed-Object Multislice Ptychography Fig. 1 | Principles of numerical diffraction pattern formation in ptychography. a Schematic illustration of the multislice 4D-STEM simulation for a thick sample. First, the beam propagation is split into multiple intervals (5 in the depicted case). The projected electrostatic potential from each of the intervals (slices) constitutes the phase of the corresponding transmission function Oi. The exit wavefront ψout is obtained via sequential application of the transmission functions Oi and Fresnel propagators P Fresnel to the incoming wavefront ψin55. The diffraction pattern is calculated as the squared modulus of the Fourier transformed exit wave. In order to account for various types of incoherent scattering, the diffraction pattern is obtained as the sum over the scattered intensities from different configurations in the detector plane. Two types of variations in configuration are depicted in (b, c). b The effect of finite electron source size, creating a superposed ensemble of shifted beams where the occurrence probability of each beam is approximated to obey a Gaussian distribution. c, Principle of thermal diffuse scattering (TDS) caused by atomic movement. The diffraction patterns corresponding to quasi-static sample configurations14,17,35 at multiple points in time ti are incoherently summed to obtain an expected intensity of a diffraction pattern with diffuse background. d Electron ptychography reconstruction models with increasing complexity levels: 2D projected phase reconstruction, 3D multislice ptychography and our proposed reconstruction method—4D mixed-object multislice ptychography, where the sequence of object states can be transformed to obtain correlations in variations of interatomic distances, schematically depicted as springs. from 2D ptychography19 to 3D multislice ptychography10,20 to CAVIAR is shown in Fig. 1d. With the retrieval of information about correlated movements of atoms (at atomic resolution), mixed-object ptychography becomes a complementary technique to vibrational electron energyloss spectroscopy (EELS) in scanning transmission electron microscopy (STEM)21,22. Vibrational STEM-EELS has been very successful in studying vibrationa (...truncated)