plasmon-phonon-polaritons in encapsulated phosphorene

We consider a system consists of a doped monolayer phosphorene embedded between two hexagonal boron nitride (hbn) slabs along the heterostructure direction. the wavevector azimuthal angle dependence of the plasmon-polariton and plasmon-phonon-polariton modes of the hybrid system are calculated based on the random-phase approximation at finite temperature. the collective modes illustrate strong anisotropy and strong coupling with phonon modes of the polar media, and furthermore, the landau damping occurs due to the intraband processes when plasmon enters intraband electron-hole continuum. our numerical results show that the plasmon mode is highly confined to the surface along the zigzag direction. owing to the strong electron-phonon interaction, the phonon dispersions in the reststrahlen bands are also angle-dependent. these results are also in agreement with those of the semiclassical model obtained in our calculations.