  We explore the temperature-dependent plasmonic modes of an n-doped
double-layer silicene system which is composed of two spatially separated
single layers of silicene with a distance large enough to prevent the
interlayer electron tunneling. By applying an externally applied electric
field, we numerically obtain the poles of the loss function within the
so-called random phase approximation, to investigate the effects of temperature
and geometry on the plasmon branches in three different regimes: topological
insulator, valley-spin polarized metal, and band insulator. Also, we present
the finite-temperature numerical results along with the zero-temperature
analytical ones to support a discussion of the distinct effects of the external
electric field and temperature on the plasmon dispersion. Our results show that
at zero temperature both the acoustic and optical modes decrease by increasing
the applied electric field and experience a discontinuity at the valley-spin
polarized metal phase as the system transitions from a topological insulator to
a band insulator. At finite temperature, the optical plasmons are damped around
this discontinuity and the acoustic modes may exhibit a continuous transition.
Moreover, while the optical branch of plasmons changes non-monotonically and
noticeably with temperature, the acoustic branch dispersion displays a
negligible growth with temperature for all phases of silicene. Furthermore, our
finite-temperature results indicate that the dependency of two plasmonic
branches on the interlayer separation is not affected by temperature at long
wavelengths; the acoustic mode energy varies slightly with increasing the
interlayer distance, whereas the optical mode remains unchanged.
