We present a theoretical investigation of quantum transport in germanene, a topological insulator, under the influence of a perpendicular (z-direction) magnetic field and circularly polarized light. Using the Kubo formalism, we compute the Hall and longitudinal conductivities and demonstrate well-defined Landau quantization together with pronounced Shubnikov de Haas oscillations. The light polarization \( l = \pm 1 \) induces opposite Floquet driven shifts in the Landau level spectrum of the two valleys, leading to asymmetric Hall plateaus and small phase shifts in the longitudinal conductivity at the single valley level. When both valleys are included, these opposite valley dependent shifts cancel, restoring the expected sequence of quantized Hall plateaus. The longitudinal conductivity exhibits nearly identical Shubnikov de Haas phase shifts for electrons and holes, indicating that the Floquet induced renormalization of the Dirac mass operates in an almost electron–hole symmetry manner. Our findings highlight the essential interplay between Floquet engineering, spin–orbit coupling, and valley degrees of freedom, and demonstrate that circularly polarized light can effectively tune magneto-transport characteristics in germanene without destroying their quantized structure.
