Abstract
The optical nonlinearity of Dirac fermions in two and three dimensions show unique properties determined by analytic expressions in the independent particle approximation, which promotes their host materials as potential candidates for providing optical nonlinear functionalities. In this work we theoretically study the third-order optical nonlinearity for one-dimensional Dirac fermions and obtain analytical expressions for third-order optical conductivities at general light frequencies. The conductivity includes many resonant peaks induced by the resonant optical transitions associated with intraband motion as well as one-, two-, and three-photon interband processes. The conductivities for field-induced second harmonic generation and third harmonic generation are discussed in detail. To connect with real materials, we choose armchair graphene nanoribbons, which host one-dimensional Dirac fermions for its low energy electronic excitation, and our analytical expressions are applied to get the third-order conductivity for harmonic generations with different ribbon width. By comparing with numerical results evaluated from a full band structure in a tight binding model, the analytic and numerical results agree pretty well for photon energy below 2.5 eV.
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