Chiral CNTs

When we compare the calculated Raman intensities for armchair, zigzag and chiral CNTs of similar diameters, we do not see large differences in the lower frequency Raman modes. This is because the lower frequency modes have a long... 

Haase and co-workers investigated electro-optic and dielectric properties of ferroelectric liquid crystals doped with chiral CNTs [495, 496]. The performance of the doped liquid crystal mixture was greatly affected even by a small concentration of CNTs. The experimental results were explained by two effects (1) the spontaneous polarization of the ferroelectric liquid crystal is screened by the 7t-electron system of the CNT and (2) the CNT 7i-electrons trap ionic impurities, resulting in a significant modification of the internal electric field within liquid crystal test cells. 

Within the formalism of the irreducible representation the position operator is decomposed in qFi + o o for chiral CNTs and E +o o for achiral topology. 

Let start to analyze the element for chiral CNT. A x,y,z reference frame intrinsic to the CNT is introduce with the z axis along the tube axis. For chiral CNT, V is the only additional symmetry operation, which is equivalent to the inversion of z axis. An electronic wavefunction labelled with + corresponds to a distribution invariant under U, and is then characterized by n = +1. On the other hand a wavefunction with change its sign under U and then has n = — 1. Because of z has n = -1, it turns out that, the third selection rules in Eq. (11) asserts that for chiral CNT we have the following non-vanishing matrix elements... 

The form of the (3 tensor for chiral CNT can be established quite generally from consideration based on the symmetry group G. As described previously, chiral CNT are invariant under U symmetry transformation. If the x axis is taken along U, the transformation corresponds to x -> x, y -> —y, z -c. It follows that, under V. we have l zzy Pzzy Pyyz Pyyz ... 

The hyperpolarizability of a chiral CNT is obtained by integrating after summation over band index m, for k inside the ID and adding the k = Q contribution... 

Within Are model considered and in the limit of the described approximations, the code estimates a value (3 10" ° esu for most of the chiral CNT topologies. No regular dependence of f3 on CNT diameter, ranging from 7.7 to 19.68 A for the investigated topologies, has been found in simulations. 

T varies from 235 fs for E — Ep = 0.03 eV to 18 fs for E — Ep > 1.5 eV. It results that, for excitation wavelengths in the near infrared (w 1 oVf the hyperpolarizability can be enhanced up to one order of magnitude under resonant interaction regime. HyperpolarizabiUties of the order of 10 esu are then in principle possible for selected chiral CNT topologies. This value is slightly lower then the estimated hyperpolarizability of chromophores and Sol-Gel, usually accounted as the most efficient second order nonlinear optical materials [29]. 

Fig. 9. The Il-orbital (tight-binding approximation) EB spectrum of the (8,2)-chiral CNT. Note the independent symmetries of the EBs with respect io —k and Ei - -Ei.

A much more complete model of ultrashort chiral CNTs was synthesized later that year by Isobe et al. [39]. While not, strictly speaking, a cycloparaphenylene, these cyclocrysenylenes (CC) exhibit atropisomerism, with rotation of each chrysenylene unit through the [4]CC cavity geometrically disfavored. These macrocycles were synthesized with an organometallic approach, similar to that used by Yamago (Fig. 39). 

SWCNTs can be considered as rectangular strips of hexagonal graphite monolayers rolling up to cylinder tubes. Two types of SWCNTs with high symmetry are normally selected by researchers, which are zigzag SWCNTs and armchair SWCNTs. When some of the atomic bonds are parallel to the tube axis, the CNT is called a zigzag CNT, while if the bonds are perpendicular to the axis, it is called an armchair CNT, and for any other structures, they are called chiral CNTs, as shown in Fig. 16.6 [72]. 

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