Valley-controlled transport in graphene/ WSe$_{2}$ heterostructures under an off-resonant polarized light

We investigate the electronic dispersion and transport properties of graphene/WSe$_{2}$ heterostructures in the presence of a proximity-induced spin-orbit coupling $\lambda_{v}$, sublattice potential $\Delta$, and an off-resonant circularly polarized light of frequency $\Omega$ that renormalizes $\Delta$ to $\bar{\Delta}_{\eta p} = \Delta +\eta p \Delta_{\Omega}$ with $\eta$ and $p$ the valley and polarization indices, respectively, and $\Delta_{\Omega}$ the gap due to the off-resonant circularly polarized light. Using a low-energy Hamiltonian we find that the interplay between different perturbation terms leads to inverted spin-orbit coupled bands. At high $\Omega$ we study the band structure and dc transport using the Floquet theory and linear response formalism, respectively. We find that the inverted band structure transfers into the direct band one when the off-resonant light is present. The valley-Hall conductivity behaves as an even function of the Fermi energy in the presence and absence of this light. At $\Delta_{\Omega}$ = $\lambda_{v}$ - $\Delta$ a transition occurs from the valley-Hall phase to the anomalous Hall phase. In addition, the valley-Hall conductivity switches sign when the polarization of the off-resonant light changes. The valley polarization vanishes for $\Delta_{\Omega}$ = 0 but it is finite for $\Delta_{\Omega}$ $\neq$ 0 and reflects the lifting of the valley degeneracy of the energy levels, for $\Delta_{\Omega} \neq 0$, when the off-resonant light is present. The corresponding spin polarization, present for $\Delta_{\Omega}$ = 0, increases for $\Delta_{\Omega}$ $\neq$ 0. Further, pure $K$ or $K^{\prime}$ valley polarization is generated when $\Delta_{\Omega}$ changes sign. Also, the charge Hall conductivity is finite for $\Delta_{\Omega}\neq 0$ and changes sign when the handedness of the light polarization changes.