Linear optical properties of deformed carbon nanotubes

Optical absorptions of the deformed single-walled carbon nanotubes (SWNTs) have been studied by the $\pi$-band tight-binding calculation, in which the matrix elements of the momentum operator are evaluated in an analytical way similar to the first-principles calculation. Our study shows that the linear optical properties of the deformed SWNTs are very sensitive to the kind of strains and the nanotube geometry (radius and chiral angle). It is also found that the absorption spectra of a deformed finite-length SWNT exhibit some new characters, e.g., many new peaks appear along with the original characteristic peaks. Based upon the obtained results, an effective experimental tool is proposed to easily measure the deformation degree of a deformed SWNT.

Figure 1

Geometrical structure of a carbon nanotube. ${\mathbf{r}}_1$, ${\mathbf{r}}_2$, and ${\mathbf{r}}_3$ correspond to bond 1, 2, and 3, respectively. ${\stackrel{⃑}{a}}_1$ and ${\stackrel{⃑}{a}}_2$ are the lattice vectors of the two-dimensional sheet. $\widehat{c}$ and $\widehat{t}$ are unit vectors along the circumference and tube axis, respectively.

Figure 2

Real part of the parallel conductivity ${\sigma }_{\text{parallel}}^R$ (left panel) and the corresponding DOS (right panel) of the zigzag (20, 0) tube with increasing uniaxial strain.

Figure 3

Real part of the parallel conductivity ${\sigma }_{\text{parallel}}^R$ (left panel) and the corresponding DOSs (right panel) of the armchair (10, 10) tube with increasing torsional strain. $\gamma$ is the shear strain.

Figure 4

Change of $N=9$ energy band for the tube (10, 10) under torsional strain. The solid and dotted line correspond to 0° and 2° strains, respectively. Here, $k$ is in unit of $1∕a_0$.

Figure 5

The first optical absorption peak positions versus applied strains, calculated by Eq. (5). (a) Zigzag (20, 0) tube under larger $e_t$ and (b) armchair (10, 10) tube under larger $\gamma$. The analytical results under small strains obtained by Eq. (10) are shown in (c) zigzag (20, 0) tube under uniaxial strain $e_t$ and (d) armchair (10, 10) tube under torsional strain $\gamma$.

Figure 6

(a) Real part of the parallel conductivity ${\sigma }_{\text{parallel}}^R$ and (b) DOS, for the finite tube (10, 10) with the tube length of 70 periods under torsional strain $(\gamma =2°)$.