  The advent of monochromated electron energy-loss spectroscopy has enabled
atomic-resolution vibrational spectroscopy, which triggered interest in
spatially localized or quasi-localized vibrational modes in materials. Here we
report the discovery of phonon vortices at heavy impurities in two-dimensional
materials. We use density-functional-theory calculations for two configurations
of Si impurities in graphene, Si-C3 and Si-C4, to examine atom-projected phonon
densities of states and display the atomic-displacement patterns for select
modes that are dominated by impurity displacements. The vortices are driven by
large displacements of the impurities, and reflect local symmetries. Similar
vortices are found at phosphorus impurities in hexagonal boron nitride,
suggesting that they may be a feature of heavy impurities in crystalline
materials. Phonon vortices at defects are expected to play a role in thermal
conductivity and other properties.
