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Atomic-scale imaging of frequency-dependent phonon anisotropy

Nature volume 645, pages 893–899 (2025)Cite this article

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Abstract

Directly visualizing vibrational anisotropy in individual phonon modes is essential for understanding a wide range of intriguing optical, thermal and elastic phenomena in materials1,2,3,4,5. Although conventional optical and diffraction techniques have been used to estimate vibrational anisotropies, they fall short in achieving the spatial and energy resolution necessary to provide detailed information4,5,6,7. Here, we introduce a new form of momentum-selective electron energy-loss spectroscopy, which enables the element-resolved imaging of frequency- and symmetry-dependent vibrational anisotropies with atomic resolution. Vibrational anisotropies manifest in different norms of orthogonal atomic displacements, known as thermal ellipsoids. Using the centrosymmetric strontium titanate as a model system, we observed two distinct types of oxygen vibrations with contrasting anisotropies: oblate thermal ellipsoids below 60 meV and prolate ones above 60 meV. In non-centrosymmetric barium titanate, our approach can detect subtle distortions of the oxygen octahedra by observing the unexpected modulation of q-selective signals between apical and equatorial oxygen sites near 55 meV, which originates from reduced crystal symmetry and may also be linked to ferroelectric polarization. These observations are quantitatively supported by theoretical modelling, which demonstrates the reliability of our approach. The measured frequency-dependent vibrational anisotropies shed new light on the dielectric and thermal behaviours governed by acoustic and optical phonons. The ability to visualize phonon eigenvectors at specific crystallographic sites with unprecedented spatial and energy resolution opens new avenues for exploring dielectric, optical, thermal and superconducting properties.

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Fig. 1: Acquisition of vibrational spectra of SrTiO3 and BaTiO3 films.
Fig. 2: Atomic-resolution q-selective EELS signal mapping of STO in different energy ranges with two orthogonal displacement directions of EEA.
Fig. 3: Atomic-resolution q-selective EELS signal mapping of BTO.
Fig. 4: Quantitative analysis of vibrational anisotropies of oxygen atoms in STO and BTO in different energy ranges.
Fig. 5: Simulated frequency-dependent thermal ellipsoids of oxygen atoms in STO and BTO.

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Data availability

The datasets generated and analysed during the current study are available from the corresponding authors upon request.

Code availability

The MATLAB code for the EELS data processing can be found on GitHub at https://github.com/PanGroup-UCI/Vibrational-EELS_background_subtraction.

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Acknowledgements

The experimental work was primarily supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering (Grant No. DE-SC0014430). Further support was provided by the National Science Foundation through a Materials Research Science and Engineering Center programme (Grant Award No. DMR-2011967). We acknowledge the use of facilities and instrumentation at the University of California (UC), Irvine’s Materials Research Institute (IMRI). J.R. and P.M.Z. acknowledge the Swedish Research Council (Grant No. 2021-03848), Olle Engkvist’s foundation (Grant No. 214-0331), Carl Trygger’s Foundation, the Knut and Alice Wallenberg Foundation (Grant No. 2022.0079), and STINT’s Joint Sweden–China Mobility programme (Grant No. CH2019-8211) for financial support. Simulations were enabled by resources provided by the National Academic Infrastructure for Supercomputing in Sweden and the Swedish National Infrastructure for Computing at the NSC Centre, which was partially funded by the Swedish Research Council (Grant Agreement Nos. 2022-06725 and 2018-05973). This work used the infrastructure for high-performance and high-throughput computing, research data storage and analysis, and scientific software tool integration built, operated and updated by the Research Cyberinfrastructure Center at UC, Irvine. The Research Cyberinfrastructure Center provides cluster-based systems, application software and scalable storage to directly support the UC, Irvine research community. We thank M. Xu and S. J. Kim (UC, Irvine) for the wet transfer of free-standing films, T. C. Lovejoy and N. Dellby (Bruker AXS LLC) for microscope alignment and high-order EELS aberration correction, and B. Liao (UC, Santa Barbara) for valuable discussions.

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Author notes

  1. These authors contributed equally: Xingxu Yan, Paul M. Zeiger, Yifeng Huang

Authors and Affiliations

  1. Department of Materials Science and Engineering, University of California, Irvine, CA, USA

    Xingxu Yan, Hongbin Yang & Xiaoqing Pan

  2. Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden

    Paul M. Zeiger & Ján Rusz

  3. Department of Physics and Astronomy, University of California, Irvine, CA, USA

    Yifeng Huang, Jie Li, Chaitanya A. Gadre, Ruqian Wu & Xiaoqing Pan

  4. National Laboratory of Solid State Microstructures, Jiangsu Key Laboratory of Artificial Functional Materials, College of Engineering and Applied Sciences, Nanjing University, Nanjing, China

    Haoying Sun & Yuefeng Nie

  5. Key Laboratory of Magnetic Materials Devices & Zhejiang Province Key Laboratory of Magnetic Materials and Application Technology, Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo, China

    Ri He & Zhicheng Zhong

  6. Irvine Materials Research Institute, University of California, Irvine, CA, USA

    Toshihiro Aoki & Xiaoqing Pan

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  1. Xingxu Yan
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Contributions

X.P. conceived this project and designed the studies, with contributions from J.R., R.W. and X.Y. X.Y. and Y.H. performed the STEM-EELS experiments and analysed all datasets with the help of H.Y., C.A.G. and T.A. Y.H. performed the drift correction. P.M.Z. and J.R. designed and performed the molecular dynamic-based calculation of vibrational spectra with the help of Y.H., R.H. and Z.Z. J.L. and R.W. conducted the first-principles calculations. The free-standing STO and BTO films were synthesized and provided by H.S. and Y.N. All authors discussed and commented on the results. The paper was prepared by X.Y., P.M.Z., J.R., R.W. and X.P. with contributions from all other co-authors.

Corresponding authors

Correspondence to Ruqian Wu, Ján Rusz or Xiaoqing Pan.

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Extended data figures and tables

Extended Data Fig. 1 TEM analysis of STO and BTO films supported by a few-layer graphene suspended on a lacey carbon film.

a, A low-magnification TEM image of the STO film supported by a graphene film with 3–6 layers on a lacey carbon TEM grid. All experiments were conducted on the suspended perovskite/graphene regions without the carbon film. b,c, Selected area diffraction patterns of a pure graphene region (b) and a STO/graphene region (c) as marked in (a), respectively. There are six sets of typical diffraction patterns of graphene in (b), indicating the layer number is probably six with random stacking orders. The diffraction pattern of STO along [001] direction is clearly visible in (c), while the diffraction patterns of graphene are hardly visible due to its thin thickness. d, A low-magnification TEM image of the BTO film supported by a graphene film on a lacey carbon TEM grid. e, Selected area diffraction pattern of a BTO/graphene region as marked in (d). f,g, Low- (f) and medium-magnification (g) STEM images of the STO film. h-j, Core-loss EELS analysis of elemental distribution and chemical composition of the STO film using Ti L2,3 edge signal (h), O K edge signal (i), and energy difference between t2g and eg peaks in the Ti L3 edge (j). k,l, Low- (k) and medium-magnification (l) STEM images of the BTO film. m-o, Core-loss EELS analysis of elemental distribution and chemical composition of the BTO film using Ti L2,3 edge signal (m), O K edge signal (n), and energy difference between t2g and eg peaks in the Ti L3 edge (o). Although there are dislocations in both STO and BTO as the white segments and lines marked in (f) and (k), we can easily find defect-free regions larger than 50 nm × 50 nm in both samples. All vibrational EELS hyperspectral imaging datasets were acquired from those single-crystalline regions without any visible contamination and away from dislocations and other defects as shown in (g) and (l). All investigated regions were also tilted on zone-axis as shown in (c) and (e). Thus, the collected phonon signals cannot be interfered with by defects, contamination, overall strain field or bent contours. The elemental distributions of Ti and O based on core-loss EELS signals are uniform in both samples. The splitting of t2g and eg peaks as the energy difference between these two peaks in the Ti L3 edge, which is widely used to estimate the change of valence state of Ti, is 2.44–2.46 eV in STO (j) and 2.20–2.21 eV in BTO (o), respectively. Since the oxygen vacancy and associated valence state change of Ti can lead to a reduction of the splitting for about 0.5 eV in the literature17, core-loss EELS results can exclude the occurrence of oxygen vacancies or other compositional variations in the investigated regions. Therefore, STEM images and elemental mapping results can rule out the interference from dislocations, oxygen vacancies, and composition changes in the studied areas for the phonon anisotropy results.

Extended Data Fig. 2 Background subtraction and comparison of vibrational signals between STO and graphene films.

a, A HAADF STEM image at the edge of STO film. The dark region is graphene, while the bright region is STO supported by graphene. b, Raw dark-field vibrational spectra at STO and graphene regions as marked by black and red dots in (a). Both spectra were obtained by summing 200 frames of 1 s exposure spectra. The spectrum collected at the graphene region contains two broad peaks at 60–100 and 140–200 meV, in agreement with previous studies11,12. c, Background-subtracted spectra at STO and graphene regions. In the energy range of 10–80 meV energy range, the vibrational signals of graphene are negligible compared to those of STO. In the energy range of 80–120 meV, the intensity of vibrational signals of STO is about six times that of graphene. The broad peak in 140–200 meV originates from phonon modes in graphene or other residual contamination. Therefore, we can ignore the bulk phonon modes of graphene and residual contamination in the energy range of interest for STO and BTO. d, Background-subtracted spectrum containing STO phonon structure in the energy range of 10–110 meV. The spectrum is used in Fig. 1b. e, Background-subtracted spectrum containing vibrational states of graphene and residual contamination in the energy range of 140–200 meV. The raw spectrum is duplicated from the black curve in (b). The blue dots are fitting windows, while the black dashed lines are the fitted backgrounds using a power law function.

Extended Data Fig. 3 Simulated phonon dispersion curves, PhDOS, and vibrational spectra of STO and BTO using different methods.

a, Density functional perturbation theory (DFPT)-simulated phonon dispersion curves of STO at 0 K. b, PhDOS and atom projected PhDOS associated with (a). c, Phonon dispersion curves of STO simulated from the DeePMD potential at 0 K using the finite displacement method. d, PhDOS and atom projected PhDOS associated with (c). e, Phonon dispersion curves of STO simulated from the DeePMD potential at 300 K using the finite displacement method. f, PhDOS and atom projected PhDOS associated with (e). g, Phonon dispersion curves of BTO simulated from the DeePMD potential at 300 K using the finite displacement method. h, PhDOS and atom projected PhDOS associated with (g). i, Modified PhDOS of STO and atom projected PhDOS with a 7 meV Gaussian broadening after involving phonon occupation number and phonon energy. j, Simulated vibrational EEL spectrum of STO with a 13 meV Gaussian broadening after subtracting the background. k, Modified PhDOS of BTO and atom projected PhDOS with a 7 meV Gaussian broadening after involving phonon occupation number and phonon energy. l, Simulated vibrational EEL spectrum of BTO with a 13 meV Gaussian broadening after subtracting the background. In (b, d, f, i, j), green, blue, and red curves represent the projected PhDOS of Sr, Ti, and O atoms, respectively. In (h, k, l), orange, blue, and red curves represent the projected PhDOS of Ba, Ti, and O atoms, respectively. LA/TA modes and four pairs of LO/TO modes are labeled at the Γ point for each phonon dispersion curve. Only three pairs of LO/TO modes are infrared active, but all should be visible in vibrational EELS results due to their different selection rules and the use of a large convergence semi-angle in our experiments. Five vertical dashed lines indicate the experimental peak centers of P1–P5 in STO at 14.6 meV, 24.4 meV, 41.1 meV, 63.3 meV, and 98.7 meV, respectively. At 0 K, both DFPT- and DeePMD-based phonon dispersion curves show obvious imaginary modes at the R and M points due to the octahedral rotation instabilities and at the Γ point due to polar instability37. Consequently, non-zero signals are present in the negative energy ranges of corresponding PhDOS curves. At 300 K, the absence of imaginary modes indicates the dynamic stability of such a structure at room temperature. All three simulated PhDOS curves reasonably match experimental peak centers, despite some discrepancies in energy positions. Among them, DeePMD-simulated PhDOS at 300 K agrees best with experimental peak values, especially for P1–P4. The overall phonon band structure of BTO is similar to that of STO but with noticeable red-shifts in the high-energy O-related peaks (P4 and P5) and blue-shifts in the Ti-O related peaks (P2 and P3). To make a more coherent comparison, we multiplied the simulated PhDOS with (n + 1)/ ω, where n is the phonon occupation number obeying the Boltzmann distribution and ω is the phonon energy, to convert to vibrational EELS signals. After the modification, the high-energy phonon modes above 30 meV (P3, P4, and P5) in (i) and (k) are significantly suppressed and look more consistent with those in experimental spectra. We also provide simulated vibrational spectra using our FRFPMS method in (j) and (l), showing even better consistency with experimental spectra in terms of the intensity ratio of P1 and P2 in the low energy ranges. The remaining differences are probably due to the finite size of simulation supercell and associated issues of LO/TO splitting in periodic supercells.

Extended Data Fig. 4 Procedure of q-selective EELS data collection.

a, Low magnification STEM image before EELS acquisition. The central region with a size of 2 nm × 2 nm (the yellow box) was used to acquire an atomic resolution q-selective EELS hyperspectral image dataset. b, High magnification STEM image acquired at the yellow box in (a) with monochromation before EELS acquisition. c, Hyperspectral image dataset with 60×60 pixel area. The displayed intensity is the integrated raw signal in the 10–20 meV range. The total acquisition time for this dataset is about 60 min. d, High magnification STEM image acquired at the yellow box in (e) with monochromation after EELS acquisition. e, Low magnification STEM image after EELS acquisition. The atomic structure of STO overlaps on the atomic resolution STEM images in (b) and (d). The region of interest is chosen with perfect STO crystalline structure in absence of any defects. The crystalline structure is still retained after the long EELS acquisition. No detectable contamination occurs during this experiment as shown in (e). The red dashed circle indicates a defective region to be used to measure the sample drift before and after the EELS acquisition. The sample drift was 0.20 nm in this case with a sample drift rate of about 0.2 nm/h. f, Sequential STEM images captured during multi-frame EELS acquisition on STO of 3 nm × 3 nm. The frame number and time are labeled on top of each image with an acquisition time of 200 s per frame. A single-unit-cell atomic structure of STO is overlapped on atomic resolution STEM images, and the green, cyan, black, purple, and orange dots denote Sr, Ti, O1, O2 (apical oxygen), and O3 (equatorial oxygen), respectively.

Extended Data Fig. 5 Data analysis, drift correction, and the influence of energy resolution.

a, Background subtracted vibrational spectra in one hyperspectral image dataset. b, Affine transformation of the raw map in 10–20 meV to reconstruct the square lattice of STO by first finding the center of bright spots at Sr positions and applying a series of affine transformations for each unit cell. c, Affine transformation of the raw map in the 10–20 meV energy range to reconstruct the square lattice of STO by using the same transformation matrix H measured in individual unit cells in (b). The selected unit cells are indicated as parallelograms and then reconstructed to squares in (b) and (c). d,e, Representative vibrational spectra of STO (d) and BTO (e) containing ZLPs with a full-width at half-maximum of 11.3 meV and 12.2 meV, respectively. f,g, Distribution of energy resolution of all datasets on STO (f) and BTO (g). We made statistical analysis of all experimental datasets of both STO and BTO and found that the average energy resolution of STO datasets (11.1 ± 0.7 meV) was smaller than that of BTO (12.9 ± 0.5 meV).

Extended Data Fig. 6 Atomic resolution vibrational signal maps with single unit cell regions and raw simulation results.

a,b, Experimental vibrational signal maps of STO in different energy ranges with DEEA = (−62, 0) or an X’ shift (a), and with DEEA = (0, −62) or a Y’ shift (b) along with aHAADF images and atomic structures, obtained from the same datasets in Fig. 2. c,d, Experimental vibrational signal maps of BTO in different energy ranges with DEEA = (−62, 0) or an X’ shift (a), and with DEEA = (0, −62) or a Y’ shift (b) along with aHAADF images and atomic structures, obtained from the same datasets in Fig. 2. The green, gold, cyan, black, purple, and orange dots denote Sr, Ba, Ti, O1, O2 (apical oxygen), and O3 (equatorial oxygen), respectively. Scale bars are 2 Å. e–h, Raw simulated energy-filtered vibrational signal maps from 1 THz and 25 THz of STO for the X’ shift (e), STO for the Y’ shift (f), BTO for the X’ shift (g), and BTO for the Y’ shift (h). The size of each image is 8 Å × 8 Å. The energy bin width is 1 THz, and the labeled energy is the bin center. The atomic structure overlaps on the first signal map of each situation as A-site (Sr or Ba) atoms residing at the top-left corner of each image.

Extended Data Fig. 7 More atomic resolution vibrational signal mapping of STO and BTO over larger regions.

a,b, Experimental vibrational signal maps of STO on a 2.3 nm × 2.3 nm region in different energy ranges with DEEA = (−62, 0) or an X’ shift (a), and with DEEA = (0, −62) or a Y’ shift (b). c, Vibrational signal maps averaged from 20 unit cells in (a). d, Vibrational signal maps averaged from 16 unit cells in (b). Displayed maps in (c) and (d) are duplicated from averaged single-unit-cell results to 2×2 unit cells for visual clarity. e,f, Experimental vibrational signal maps of BTO on a 2.3 nm × 2.3 nm region in different energy ranges with DEEA = (−62, 0) or an X’ shift (e), and with DEEA = (0, −62) or a Y’ shift (f). An atomic structure of STO or BTO is overlapped on all vibrational signal maps, and the green, gold, cyan, black, purple, and orange circles denote Sr, Ba, Ti, O1, O2 (apical oxygen), and O3 (equatorial oxygen), respectively. g,h, Vibrational signal maps of STO (e) and BTO (f) with a smaller energy bin size of 2 meV. The dashed lines indicate the boundary between A-site dominant signal maps and Ti dominant ones. In (h), Ba columns are still observable in the signal maps of 20–22 meV, 22–24, and 24–26 meV, while Ti/O1 columns exhibit the prominent intensity only when the energy range is above 24–26 meV. By contrast, the vibrational intensity of Sr columns in STO becomes very weak when the energy range is above 18–20 meV, in parallel with increasingly visible Ti/O1 columns in (g). Therefore, we indeed detected noticeable vibrational intensity at Ba columns in 20–30 meV due to its non-zero atom-projected PhDOS and spectral broadening.

Extended Data Fig. 8 Simulated position-dependent energy-filtered diffraction patterns of both inelastically and elastically scattered electrons of STO along [001] (upper) and BTO along [010] (lower) at selected energies.

49.6 meV (12 THz) and 103.4 meV (25 THz) are selected for STO, while 53.8 meV (13 THz) and 95.1 meV (23 THz) are selected for BTO. The energy bin width is 1 THz. The atomic positions (= O3, O2, Sr or Ba, and Ti/O1) being placed by the electron beam and the energy values are labeled at the top of diffraction patterns. All inelastic signals with certain phonon energy losses (left three columns) are on a linear scale, whereas all elastic signals (right two columns) are on a logarithmic scale. Two dashed green circles indicate EEA positions of DEEA = (−62, 0) and DEEA = (0, −62), respectively. Scale bars are all 20 mrad.

Extended Data Fig. 9 Three-dimensional maps of simulated ADPs of all atoms in STO and BTO at selected energies.

(a) Maps of ADPs for STO in oblique views (upper) and top views along [001] (lower). The thermal ellipsoids for the total PhDOS (the left most image) are drawn such that each atom is found with a probability of 99% within the volume of the ellipsoid using the VESTA software58. The shape of thermal ellipsoids represents the isotropy or anisotropy of atomic displacements. In all other images to the right, the frequency-dependent ADPs of all atoms are rescaled to highlight the relative sizes of displacements at the corresponding energy. The rescaling factors for these images are approximately 264 (3 THz), 943 (6 THz), 5642 (12 THz), 5087 (14 THz), 2394 (15 THz), 5495 (16 THz), 15505 (24 THz), and 41480 (25 THz). The electron beam in STO experiments was parallel to the Z axis or [001] direction. In the map of total PhDOS to the left, the thermal vibrations of Sr and Ti atoms are isotropic, but oxygen atoms exhibit an anisotropy consistent with ref. 5. The energy-dependent displacement ellipsoids, shown to the right, extend this consideration: Sr and Ti atoms exhibit isotropic displacements at all energies, indicating the homogeneity of corresponding phonon eigenvectors, and the anisotropy of oxygen vibrations evolves as a function of frequency. Oxygen displacements form an oblate ellipsoid (U11/U22 < 1) lying on {100} planes in the lower energy range. The ellipsoids associated with oxygen atoms transform to prolate ellipsoids (U11/U22 > 1) elongated along <100> directions at higher energies. (b) Maps of ADPs for BTO in oblique views (upper) and top views along [010] (lower). In all other images to the right, the frequency-dependent ADPs of all atoms are rescaled to highlight the relative sizes of displacements at the corresponding energy. The rescaling factors for these images are approximately 31 (3 THz), 122 (4 THz), 187 (5 THz), 70 (6 THz), 130 (8 THz), 316 (12 THz), 292 (13 THz), 350 (14 THz), and 5912 (23 THz). The electron beam in BTO experiments was parallel to the Z axis or [010] direction. Note that the thermal ellipsoids of both Ba and Ti are no longer isotropic in BTO.

Extended Data Table 1 Comparison of experimental and calculated phonon energies of STO and BTO from the literature and our work
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Yan, X., Zeiger, P.M., Huang, Y. et al. Atomic-scale imaging of frequency-dependent phonon anisotropy. Nature 645, 893–899 (2025). https://doi.org/10.1038/s41586-025-09511-z

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