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 → 0) modes for the case of homopolar covalent crystals. We demonstrate that the near Γ-point frequencies of the IR-active modes are substantially damped by the inclusion of the induced dipole interaction, in agreement with experiment. The fitting procedure outlined here allows for this model to be adapted to other materials, including but not limited to two-dimensional crystals.