AbstractDirectly mapping local phonon dispersion in individual nanostructures can advance our understanding of their thermal, optical, and mechanical properties. However, this requires high detection sensitivity and combined spatial, energy and momentum resolutions, thus has been elusive. Here, we demonstrate a four-dimensional electron energy loss spectroscopy technique, and present position-dependent phonon dispersion measurements in individual boron nitride nanotubes. By scanning the electron beam in real space while monitoring both the energy loss and the momentum transfer, we are able to reveal position- and momentum-dependent lattice vibrations at nanometer scale. Our measurements show that the phonon dispersion of multi-walled nanotubes is locally close to hexagonal-boron nitride crystals. Interestingly, acoustic phonons are sensitive to defect scattering, while optical modes are insensitive to small voids. This work not only provides insights into vibrational properties of boron nitride nanotubes, but also demonstrates potential of the developed technique in nanoscale phonon dispersion measurements.
Operator definition and derivation of collisional energy and momentum loss in relativistic plasmas
