  We propose to Floquet-engineer Dirac cones at the surface of a
three-dimensional topological insulator. We show that a large tunability of the
Fermi velocity can be achieved as a function of the polarization, direction and
amplitude of the driving field. Using this external control, the Dirac cones in
the quasienergy spectrum may become elliptic or massive, in accordance to
experimental evidences. These results help us to understand the interplay of
surface states and external ac driving fields in topological insulators. In our
work we use the full Hamiltonian for the three-dimensional system instead of
effective surface Hamiltonians, which are usually considered in the literature.
Our findings show that the Dirac cones in the quasienergy spectrum remain
robust even in the presence of bulk states and, therefore, they validate the
usage of effective surface Hamiltonians to explore the properties of
Floquet-driven topological boundaries. Furthermore, our model allows us to
introduce new out-of-plane field configurations, which cannot be accounted for
by effective surface Hamiltonians.
