Franz-Keldysh effect and electric field-induced second harmonic generation in graphene: From one-dimensional nanoribbons to two-dimensional sheet

It is well established that external electrostatic fields modify the electronic structure and optical response of materials. Modifications of the optical response of quasi-one-dimensional graphene nanoribbons (GNRs) depend strongly on the direction of the electrostatic field. While transverse fields primarily lift degeneracies in the band structure, longitudinal fields are responsible for considerable nonperturbative Franz-Keldysh effects. Also, electric fields break the inversion symmetry of GNRs and result in strong electric field-induced second harmonic generation. In this work, we study field-induced modifications of the linear and nonlinear optical response of narrow and wide semiconducting armchair GNRs (AGNRs). Both finite and infinite AGNRs with and without electrostatic fields are studied and length convergence is analyzed. Similarly, the width convergence of wide AGNRs to the two-dimensional graphene limit with and without longitudinal fields is investigated.