  The magneto-optical longitudinal, transverse Hall and circularly-polarized
response of silicene and other materials described by a Kane-Mele Hamiltonian
are calculated. Particular attention is paid to the effects of an external
electric field and finite charge doping. The energy of interband transitions
can be tuned by varying the electric field. The onset frequency of the
absorptive peaks moves differently between the topological insulator and band
insulator regimes. This may be used to verify experimentally the existence of
the two insulating phases as well as provide a measure of the spin-orbit band
gap. The zeroth Landau level splits between four spin and valley distinct
energies resulting in valley-spin-polarized levels in the density of states.
With charge doping, transitions between these levels allow for a spin- and
valley-polarized response in the conductivity whereby charge carriers of
specific spin and valley index can be isolated by tuning the incident photon
frequency. Increasing the chemical potential is shown to redistribute spectral
weight from interband transitions to a strong low-energy intraband response.
For large chemical potential, this intraband feature is associated with the
semiclassical cyclotron resonance frequency which is shown to linearly increase
with magnetic field.
