  Hexagonal boron nitride (hBN) is a natural hyperbolic material which can also
accommodate highly dispersive surface phonon-polariton modes. In this paper, we
examine theoretically the mid-infrared optical properties of graphene-hBN
heterostructures derived from their coupled plasmon-phonon modes. We found that
the graphene plasmon couples differently with the phonons of the two
Reststrahlen bands, owing to their different hyperbolicity. This also leads to
distinctively different interaction between an external quantum emitter and the
plasmon-phonon modes in the two bands, leading to substantial modification of
its spectrum. The coupling to graphene plasmons allows for additional gate
tunability in the Purcell factor, and narrow dips in its emission spectra.
