Second-harmonic generation in twisted double bilayer graphene: Double-resonant enhancement from moiré flat bands

We theoretically investigate second-harmonic generation (SHG) in twisted double bilayer graphene (TDBG) with AB--AB and AB--BA stacking configurations using a perturbative approach based on an effective continuum Hamiltonian. We present a systematic analysis of the SHG response as a function of twist angle, vertical bias voltage, Fermi energy, and stacking configuration. We find that the SHG signal is strongly enhanced at small twist angles due to the emergence of moir\'{e} flat bands and the associated increase in the joint density of states. Beyond this conventional enhancement mechanism, we demonstrate that the reduced bandwidth enables a pronounced double-resonant process, in which optical transitions at both $\omega$ and $2\omega$ are simultaneously satisfied over extended regions of the moir\'{e} Brillouin zone. This mechanism leads to a substantial amplification of the SHG response, analogous to the enhancement observed in systems with discrete energy levels, but realised here in a tuneable moir\'{e} band structure. Furthermore, we show that the AB--AB and AB--BA configurations exhibit a systematic $\pi$ phase shift in the SHG response at large bias voltages, reflecting their distinct symmetry and electronic structure. Our results identify double-resonance effects as a generic mechanism for enhancing SHG in moir\'{e} systems and provide a unified framework for understanding and controlling nonlinear optical responses in tuneable flat-band materials.