Linear optical and quadratic electro-optic response of carbon nanotubes: universal analytic expressions for arbitrary chirality

Using a universal density of states (Mintmire and White 1998 Phys. Rev. Lett. 81 2506), we have found an analytic expression for the long-axis linear susceptibility of single-walled carbon nanotubes valid for arbitrary diameter and chirality. The applicability of our general expression has been assessed by comparison with numerical calculations. Excellent agreement is demonstrated in the low-energy range for semiconducting carbon nanotubes having a moderate or large diameter. The agreement is less convincing for metallic nanotubes having the same diameter as semiconducting ones and the reason for this difference has been clarified. Based on the simple closed-form expression for the linear susceptibility and using the perturbation treatment developed by Aspnes and Rowe (1972 Phys. Rev. B 5 4022), an analytic expression for the third-order nonlinear optical susceptibility χ(3)(ω;0,0,ω) has been derived for arbitrary semiconducting single-walled carbon nanotubes