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. The combination of theory and experiment provides the ability to interpret momentum-dependent phonon spectra obtained at nanometer spatial resolution in the electron microscope.