Theoretical estimations of third-order optical nonlinearities for semiconductor carbon nanotubes

In this study we present theoretical predictions concerning the third-order nonlinear optical properties of semiconductor carbon nanotubes for photon energies well below the fundamental absorption edge. Both virtual interband pi-electron transitions and combined intraband-interband ones are assumed to be the basic microscopic mechanisms of optical nonlinearities in this spectral region. Resting upon simple dimensional considerations and using only model-independent properties of the K-election energy spectrum near the conduction and valence-band edges, we obtain theoretical estimations for the low-frequency third-order susceptibility chi e((3))(0) due to these two mechanisms, which sheds light on the relationship between the non-resonant nonlinear optical response of nanotubes and their geometrical and electronic structure. This result derived in physically interpretable terms is in good agreement with that obtained in our recent study on the basis of a systematic analytical approach. We find that single-shell 'zig-zag' nanotubes display positive chi e((3))(0) values, which is due to the positive contribution from combined pi-electron transitions dominating the negative contribution from purely interband transitions. We also find that the increase of the nanotube radius R results in a strong enhancement (proportional to R-4) of chi e((3))(0), which can reach values larger by several orders of magnitude than those reported for the fullerene molecules C-60 and C-70 We draw a conclusion that the modification of the geometrical structure of nanotubes provides an efficient means for the engineering of novel nonlinear-optical materials with high cubic susceptibilities.