We present a Floquet theoretical framework to investigate the electronic and optical properties of two-dimensional hexagonal materials driven by bicircular, two-color (ω–2ω) laser fields, comprising a circularly polarized dressing field (ω) and an arbitrarily polarized driving field (2ω). The dressing field breaks time-reversal symmetry to open a bandgap at the Dirac points, while the two-color phase delay and the driving field ellipticity serve as key parameters for dynamically tuning the system response. Specifically, the driving field ellipticity induces pseudospin tilting to modulate light-matter interaction strength, whereas the phase delay reshapes the band structure by controlling the effective flux through the unit cell. As a result, co-rotating field configurations significantly enhance longitudinal photocurrents, while counter-rotating fields suppress them. Crucially, the transverse conductivity exhibits a strong dependence on the relative signs of the valley masses, enabling a robust mechanism for all-optical valley selection where one valley becomes gapless while the other remains gapped. By selectively opening or closing transverse transport channels, the phase delay and ellipticity offer versatile degrees of freedom for engineering valley-dependent optoelectronic properties in 2D hexagonal lattices.
