  We investigate the optical response of the silicene and similar materials,
such as germanene, in the presence of an electrically tunable band gap for
variable doping. The interplay of spin orbit coupling, due to the buckled
structure of these materials, and a perpendicular electric field gives rise to
a rich variety of phases: a topological or quantum spin Hall insulator, a
valley-spin-polarized metal and a band insulator. We show that the dynamical
conductivity should reveal signatures of these different phases which would
allow for their identification along with the determination of parameters such
as the spin orbit energy gap. We find an interesting feature where the electric
field tuning of the band gap might be used to switch on and off the Drude
intraband response. Furthermore, in the presence of spin-valley coupling, the
response to circularly polarized light as a function of frequency and electric
field tuning of the band gap is examined. Using right- and left-handed circular
polarization it is possible to select a particular combination of spin and
valley index. The frequency for this effect can be varied by tuning the band
gap.
