Communication: Effect of accidental mode degeneracy on Raman intensity in 2D materials: Hybrid functional study of bilayer phosphorene.

Bulk black phosphorus has two optical phonon modes labeled as Ag (2) and B2u, respectively, that are nearly degenerate in frequency. However, density functional theory calculations using local or semi-local functionals cannot reproduce this degeneracy. Here, we propose a hybrid functional approach aided by van der Waals (vdW) force fields, which can accurately describe the lattice dynamic and electronic properties of both bulk and few-layer black phosphorus (phosphorene). Using this approach we show that in bilayer phosphorene, the two Raman modes derived from the B2u and Ag (2) modes could exhibit strong resonance as a result of the accidental degeneracy so that both modes could be observed in Raman experiment. Without the mode degeneracy, however, the Raman intensity of the B2u-derived mode would be too weak to be observed. We further show that the accidental degeneracy is correlated to the applied strain, which enables Raman spectroscopy to be a powerful tool for characterizing built-in strains in 2D materials, e.g., due to the interaction with substrates, which has emerged as an important issue in vdW epitaxy.