  We study the hybrid excitations due to the coupling between surface optical
phonons of a polar insulator substrate and plasmons in the
valley-spin-polarized metal phase of silicene under an exchange field. We
perform the calculations within the generalized random-phase approximation
where the plasmon-phonon coupling is taken into account by the long-range
Fr$\ddot{\mathrm{o}}$hlich interaction. Our investigation on two hybridized
plasmon branches in different spin and valley subbands shows distinct behavior
compared to the uncoupled case. Interestingly, in one valley, it is found that
while the high energy hybrid branch is totally damped in the spin-up state, it
can be well-defined in the the spin-down state. Moreover, we show that the
electron-phonon coupling is stronger in both spin-down subbands, regardless of
valley index, due to their higher electron densities. In addition, we study the
effects of electron-phonon coupling on the quasiparticle scattering rate of
four distinct spin-valley locked subbands. The results of our calculations
predict a general enhancement in the scattering rate for all subbands, and a
jump in the case of spin-down states. This sharp increase associated to the
damping of hybrid plasmon modes is almost absent in the uncoupled case. The
results suggest an effective way for manipulating collective modes of
valley-spin-polarized silicene which may become useful in future valleytronic
and spintronic applications.
