Breakdown of optical phonons' splitting in two-dimensional materials

We investigate the long-wavelength dispersion of longitudinal and transverse optical phonon modes in pillar two-diinensional materials, mriltilayers, and their heterostructures. Using analytical models and density-functional perturbation theory in a two-dimensional framework, we show that at variance with the three-dimensional case these modes are degenerate at the zone center but the macroscopic electric field associated with the longitudinal-optical modes gives rise to a finite slope at the zone center in their corresponding phonon dispersions. This slope increases linearly with the number of layers and it is determined solely by the Born effective charges of the material and the.dielectric properties of the surrounding media. Screening from the environment can greatly reduce the slope splitting between the longitudinal and transverse optical modes arid can be seen in the experimentally relevant case of boron nitride graphene heterostructures. As the phonon momentum increases, the intrinsic screening properties of the two-dimensional material dictate the transition to a momentum-independent splitting similar to that of three-dimensional materials. These considerations are essential to understand electrical transport and optical coupling in two-dimensional systems.