  We investigate the long-wavelength dispersion of longitudinal and transverse
optical phonon modes in polar two-dimensional materials, multilayers, 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 and 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.
