  A recent theoretical work [Nature Phys., 7, 490 (2011)] has demonstrated that
external non-equilibrium perturbations may be used to convert a two-dimensional
semiconductor, initially in a topologically trivial state, into a Floquet
topological insulator. Here, we develop a non-trivial extension of these ideas
to three-dimensional systems. In this case, we show that a two-photon resonance
may provide the necessary twist needed to transform an initially unremarkable
band structure into a topological Floquet spectrum. We provide both an
intuitive, geometrical, picture of this phenomenon and also support it by an
exact solution of a realistic lattice model that upon irradiation features
single topological Dirac modes at the two-dimensional boundary of the system.
It is shown that the surface spectrum can be controlled by choosing the
polarization and frequency of the driving electromagnetic field. Specific
experimental realizations of a three-dimensional Floquet topological insulator
are proposed.
