Quantum geometry and nonlinear optical responses in rhombohedral trilayer graphene

We theoretically study the nonlinear optical response of ABC trilayer graphene with inversion symmetry broken by the application of a displacement field perpendicular to the trilayer. We show that rhombohedral trilayer graphene exhibits a large bulk photovoltaic effect arising from a DC shift-current response. The conductivity of the trilayer contains features similar to AB bilayer graphene as well as features distinct from AB bilayer graphene. The additional features of ABC trilayer graphene relative to AB bilayer graphene arise from the quantum geometric features of the electronic band structure and can be tuned by varying the displacement field. We focus on a regime of displacement field where certain band gaps close and reopen away from the charge neutrality point, leading to drastic changes in the quantum geometric structure in momentum space, a feature characteristic of the trilayer graphene band structure. These features manifest as a sign change in shift-current conductivity in a certain frequency window and can thus serve as a probe of quantum geometry.