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

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 ${\chi }^{\left(3\right)}(\omega ;0,0,\omega )$ 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.

Figure 1

Frequency dependence of ${\chi }^{\left(3\right)}(\omega ;0,0,\omega )$ for four different CNs. In panels (a) and (c), the two lowest resonances are shown and the higher resonances are illustrated in panels (b) and (d) using a different scale.

Figure 2

Frequency dependence of ${\chi }^{\left(3\right)}(\omega ;0,0,\omega )$ for (38,0) and (50,0) zigzag CNs. The rightmost part of the plots has been multiplied by a factor of 30.

Figure 3

Comparison of analytical ${\chi }^{\left(3\right)}$ and numerical $\Delta \chi$ results taking $F=0.1\mathrm{mV}∕\mathrm{\AA }$ and $\hslash \Gamma =0.1\mathrm{eV}$ for a (7,0) CN.

Figure 4

Numerically obtained $\Delta \chi (F=5.5\mathrm{mV}∕\mathrm{\AA })$ for a (7,0) CN taking $\hslash \Gamma =0.1\mathrm{eV}$. The oscillations above the band gap demonstrate that higher-order terms in the expansion of $\chi (F,\omega )$ are significant for such large fields.