Harmonic generation of graphene quantum dots in the Hartree-Fock approximation

We theoretically investigate harmonic generation in graphene quantum dots under linearly polarized optical pulses, focusing on excitonic effects. Combining the tight-binding model and the single-particle density matrix approach, we derive semiconductor Bloch equations under a static-screened Hartree-Fock approximation. This framework characterizes the electron-electron interaction through local Hartree potentials for direct Coulomb interaction and nonlocal Fock potentials for exchange interaction. Distinct configurations of Hartree and Fock terms yield various approximation methods, including the independent-particle approximation, the mean-field approximation, the random phase approximation, and the excitonic effects. We thoroughly analyze how these approximation methods affect the electronic energy levels, linear optical absorption, and nonlinear harmonic generation. Within excitonic effects, we present the dependence of harmonic generation on the geometric variations of graphene quantum dots (sizes, triangular/hexagonal shapes, and armchair/zigzag edges) and the amplitude and polarization of electric fields. Our findings show that excitonic effects significantly enhance optical responses of graphene nanostructures. For a dot ensemble formed by randomly oriented graphene quantum dots, only odd-order harmonics exist along the polarization direction of the incident light. Crucially, harmonic generation in graphene quantum dots exhibits high tunability via geometric configuration, making them promising candidates for nonlinear optical nanodevices.