Dirac fermion time-Floquet crystal: manipulating Dirac points

We demonstrate how to control the spectra and current flow of Dirac electrons in both a graphene sheet and a topological insulator (TI) by applying either two linearly polarized laser fields with frequencies ω and 2ω or a monochromatic (one-frequency) laser field together with a spatially periodic static potential (graphene/TI superlattice). Using the Floquet theory and the resonance approximation, we show that a Dirac point in the electron spectrum can be split into several Dirac points whose relative location in momentum space can be efficiently manipulated by changing the characteristics of the laser fields. In addition, the laser-field-controlled Dirac fermion band structure - a Dirac fermion time-Floquet crystal - allows the manipulation of the electron currents in graphene and topological insulators. Furthermore, the generation of dc currents of desirable intensity in a chosen direction occurs when the biharmonic laser field is applied, which can provide a straightforward experimental test of the predicted phenomena.