  Recent developments in experiments with vibrational electron energy loss
spectroscopy (EELS) have revealed spectral shape variations at spatial
resolutions down to sub-atomic scale. Interpretation in terms of local phonon
density of states enables their qualitative understanding, yet a more detailed
analysis is calling for advances in theoretical methods. In Zeiger and Rusz,
Phys. Rev. Lett. 124, 025501 (2020) we have presented a frequency resolved
frozen phonon multislice method for simulations of vibrational EELS. Detailed
simulations for a plane wave electron beam scattering on a vibrating hexagonal
boron nitride are presented in a companion manuscript (Zeiger and Rusz,
arXiv:2104.03197). Here we present simulations of vibrational EELS assuming a
convergent electron probe of nanometer size and atomic size on a hexagonal
boron nitride structure model with a planar defect. With a nanometer beam we
observe spectral shape modifications in the presence of the defect, which are
correlated with local changes of the phonon density of states. With an atomic
size electron beam, we observe the same, although with better contrast. In
addition, we observe atomic level contrast and sub-atomic scale spectral shape
modifications, which are particularly strong for small detector collection
angles.
