Mapping Phonon Dispersion Surfaces at Nanometer Scale

Due to instrument improvements in recent years, acquisition of phonon dispersions in the electron microscope has become feasible with the associated benefit of nanometer spatial resolution [1, 2]. More recently, high-fidelity phonon dispersions along a line in reciprocal space have been obtained in a few minutes, using parallel acquisition schemes with a slot aperture in the spectrometer entrance plane [3]. The acquisition of phonon dispersions of nanometer regions is of technological importance as it allows us to understand relevant material properties with sufficient spatial resolution for contemporary technology nodes. However, the experimental data and accompanying theoretical phonon dispersions are typically obtained in 1 D only: along high-symmetry momentum directions. It would be very desirable to acquire and display information in 2D and eventually in 3D, thereby yielding a more complete picture. The non-degeneracy of transverse modes in low-symmetry directions, and phononic band gaps would thus be directly shown, and this would be an important step towards accurate extraction of the full density of states. To reconstruct the 2D phonon dispersion, the microscope (Nion HERMES operated at 80 kV with Dectris ELA detector in the IRIS spectrometer) was set up for “normal” parallel phonon dispersion acquisition with a slot aperture in the diffraction plane. The sample was a flat, clean and approximately 50 nm thick lamella of diamond in [1-10] orientation. A row of <111> spots was aligned with the slot direction and the diffraction pattern was consecutively displaced over the slot aperture to obtain individual slices of 3D energy-scattering vector space (cf. Fig. 1). The data for each of the 13 diffraction pattern positions was post-processed by registration and integration of the 120 frames of 1s acquisition time (13×2 min = 26 min total acquisition time). The zero-loss lines were straightened and other processing steps applied before stacking the slices in 3D and shifting appropriately to reconstruct the diffraction plane. The result is given in Fig. 2 as an individual snapshot of the 3D object (2 momentum axes plus one energy axis), which reveals the shape of the phonon branches. Additional ongoing processing is the averaging of equivalent segments of reciprocal space and the segmentation of the branches (including interpolation of diffraction streaks). Meticulous processing and consideration of experimental parameters by means of simulation could make it possible to obtain the quantitative DOS from a well-defined region and thus yield access to material properties like the specific heat capacity that would be of importance for the design of highly integrated devices [4]. Sketch of the acquisition scheme and corresponding experimental data (log scale). In the first case the slot aperture (blue) is aligned with a row of diffraction spots. In the second case, the diffraction pattern is shifted relative to the slot. Two perspective views of the obtained phonon dispersion surfaces.