Interpretation of phonon spectroscopic data at atomic resolution in scanning transmission electron microscopy

We provide a theory for atomic resolution phonon spectroscopy in scanning transmission electron microscopy. This formulation goes beyond a recently proposed simple approach to atomic-scale phonon spectroscopy by explicitly including the dependence on probe position in the inelastic scattering cross section itself and considers the contribution to the spectrum from individual atoms. An application is made to existing experimental data that demonstrates how a direct comparison of the data with phonon densities of states projected onto individual atoms can be made and also the importance of contributions to the energy-loss spectrum on a particular atomic site from surrounding atoms due to probe spreading.