Shell model extension to the valence force field: application to single-layer black phosphorus.

We propose an extension of the traditional valence force field model to allow for the effect of electronic polarization to be included in the inter-atomic potential. Using density functional theory as a reference, this model is parameterized for the specific case of single-layer black phosphorus by fitting the phonon dispersion relation over the entire Brillouin zone. The model is designed to account for the effect of induced dipole interaction on the long-wavelength (|q[combining right harpoon above]| → 0) modes for the case of homopolar covalent crystals.