Hybrid excitation modes in a GaS monolayer: Effect of hole-phonon coupling

We theoretically investigate the coupling between surface optical phonons and holes in a system composed of a p-type doped GaS monolayer with a Mexican-hat valence band positioned on a polar substrate. We calculate the dynamical dielectric function within the random phase approximation by considering the hole-hole interaction in the presence of long-range polar Fr\"ohlich interaction at zero temperature. We obtain the hybrid hole-phonon collective excitation dispersion numerically and in the long-wavelength limit, also analytically and explore the effects of density, type of substrate and separation distance between GaS monolayer and substrate on the coupled modes. Our results clearly indicate a strong coupling between the surface optical (SO) phonons of substrate and holes at all hole densities. Furthermore, we compute the real and imaginary parts of the quasiparticle self-energy, spectral function and scattering rate by using the leading-order dynamic screening approximation (4pt0exG⁰W approximation) in both coupled and uncoupled systems. Interestingly, calculations of scattering rate reveal a distinctive feature by appearing two branches related to the inner and outer ring-shaped equal-energy surfaces. In the case of uncoupled state, an abrupt increase as a result of the plasmon emission is observed in one of the branches. However, the existence of two decay channels in coupled system leads to one (two) upward step(s) in the inner (outer) ring branch. Finally, we show that the scattering rate is apparently affected by the hole density and separation of p-type GaS monolayer and substrate.