Theory of momentum-resolved phonon spectroscopy in the electron microscope

We provide a theoretical framework for the prediction and interpretation of momentum dependent phonon spectra due to coherent inelastic scattering of electrons. We complete the approach with first principles lattice dynamics using periodic density functional theory and compare to recent electron energy loss measurements on cubic and hexagonal boron nitride performed within a scanning transmission electron microscope (STEM). The combination of theory and experiment provides the ability to interpret momentum dependent phonon spectra obtained at nanometer spatial resolution in the electron microscope.