Second-Order Conductivity Probes a Cascade of Singularities in a Moir\'e Superlattice

Systems lacking inversion symmetry inherently demonstrate a nonlinear electrical response (NLER) to an applied electric bias, emerging through extrinsic mechanisms. This response is highly sensitive to the electronic band structure, which can be engineered with remarkable precision in moir\'e superlattices formed from atomically thin quantum materials. Moir\'e superlattices host complex Fermi surface reconstructions near van Hove singularities (vHSs) in the electronic density of states. However, the role of these reconstructions in shaping NLER remains insufficiently understood. In this work, we systematically explore NLER in moir\'e superlattices of twisted double bilayer graphene (tDBLG) by tuning the Fermi level across multiple moir\'e bands on both sides of the charge neutrality point. We observe sharp variations and sign reversals in the NLER appearing via extrinsic pathways near mid-band vHSs. The second-order conductivity close to the vHSs demonstrates a much higher value than previous reports of extrinsic NLER in any other material. Our results demonstrate that NLER can serve as a sensitive probe of Fermi surface reconstructions and establish tDBLG as a versatile and highly efficient platform for generating and controlling the nonlinear electrical response.