Theoretical study of quadratic electro-optic effect in semiconducting zigzag carbon nanotubes

Abstract

Using the perturbation treatment developed by Aspnes and Rowe [Phys. Rev. B 5, 4022 (1972)], an analytic expression for the third-order nonlinear optical susceptibility ${\ensuremath{\chi}}^{(3)}(\ensuremath{\omega};0,0,\ensuremath{\omega})$ is computed and analyzed for single walled zigzag carbon nanotubes. By improving their method, our calculations based on a tight-binding model take into account the transitions between all pairs of valence and conduction bands and thereby the contributions to the third-order susceptibility associated with different energy bands are investigated. With increasing radius of the nanotube, a nonmonotonous increase of the quadratic electro-optic effect has been demonstrated except for the fundamental peak. The nonuniformity is a result of the overlap between two energy bands as well as the reduced effective masses associated with each pair of conduction and valence bands. A nonperturbative numerical calculation is applied to obtain the high-field response as well as to assess the applicability of the low-field perturbation expression.