  Plasmons and polar phonons are elementary electrodynamic excitations of
matter. In 2d and at long wavelengths, they couple to light and act as the
system polaritons. They also dictate the scattering of charged carriers. Van
der Waals heterostructures offer the opportunity to couple excitations from
different layers via long-range Coulomb interactions, modifying both their
dispersion and their scattering of electrons. Even when the excitations do not
couple, they are still influenced by the screening from all layers, leading to
complex dynamical interactions between electrons, plasmons and polar phonons.
We develop an efficient ab initio model to solve the dynamical electric
response of Van der Waals heterostructures, accompanied by a formalism to
extract relevant spectroscopic and transport quantities. Notably, we obtain
scattering rates for electrons of the heterostructure coupling remotely with
electrodynamic excitations. We apply those developments to BN-capped graphene,
in which polar phonons from BN couple to plasmons in graphene. We study the
nature of the coupled excitations, their dispersion and their coupling to
graphene's electrons. Regimes driven by either phonons or plasmons are
identified, as well as a truly hybrid regime at long wavelengths. Those are
studied as a function of the graphene's Fermi level and the number of BN
layers. In contrast with descriptions in terms of surface-optical phonons, we
find that the electron-phonon interaction stems from different modes. Moreover,
the dynamical screening of the coupling between BN's LO phonons and graphene's
electrons crosses over from inefficient to metal-like depending on the relative
value of the phonons' frequency and the energetic onset of interband
transitions. While the coupling is significant in general, the associated
scattering of graphene's carriers is found to be negligible in the context of
electronic transport.
