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We consider the three-dimensional Hamiltonian for Bi$_2$Se$_3$, asecond-generation topological insulator, under the effect of a periodic drivefor both in-plane and out-of-plane fields. As it will be shown by means ofhigh-frequency expansions up to second order in the Floquet Hamiltonian, thedriving induces anisotropies in the Dirac cone and opens up a quasienergy gapfor in-plane elliptically polarized fields. Analytic expressions are obtainedfor the renormalized velocities and the quasienergy gap. These expressions arethen compared to numerical calculations performed by discretizing theHamiltonian in a one-dimensional lattice and following a staggered fermionapproach, achieving a remarkable agreement. We believe our work may have animpact on the transport properties of topological insulators.

Inducing anisotropies in Dirac fermions by periodic driving