First-principles Quantum Insights into Valley Hall Effect and Nonlinear Optical Response of Ge-Doped Graphene for Potential Valleytronic Applications

Employing the first-principles density functional theoretical approach, this study first reports systematic investigation on the electronic and topological properties, and nonlinear optical phenomena of Ge-doped graphene. These include energy bands spectra, valley Hall effect, second-harmonic generation, and the Berry curvature. The substitutionally doped graphene supercells of the functional quantum materials were modeled with Ge concentrations of 12.5%, 5.5%, 3.125%, and 2%. The computed results show that Ge doping breaks the inversion symmetry of graphene and opens a tunable bandgap ranging from 0.17 eV to 1.06 eV. The calculated Berry curvature exhibits opposite signs at the K and K′ valleys, indicating valley-contrasting transportation and the emergence of a valley Hall effect. Pronounced valley polarization is observed for the 12.5% and 3.125% doped Ge-graphene quantum systems, while the 5.5% doped structure exhibits weakened valley contrast associated with a nearly closed bandgap. Spin–orbit coupling lifts band degeneracies near the valley points, whereas the calculated Chern numbers remain close to zero, confirming a topologically trivial phase. Furthermore, inversion-symmetry breaking induces strong second-order nonlinear optical responses and pronounced second-harmonic generation peaks correlated with the optical absorption spectra. These findings demonstrate that Ge doping is an effective strategy for tailoring the valleytronic and nonlinear optical properties of graphene.