Effect of bending on Raman-active vibration modes of carbon nanotubes

We investigate vibration modes and their Raman activity of single-walled carbon nanotubes that are bent within their intrinsic elastic limits. By implementing wedge boundary conditions for density-functional-based tight binding, and using nonresonant bond polarization theory, we discover that Raman activity can be induced by bending. Depending on the degree of bending, high-energy Raman peaks change their positions and intensities significantly. These effects can be explained by migration of nodes and antinodes along tube circumference. We discuss the challenge of associating the predicted spectral changes with experimental observations.

Figure 1

Schematic of the boundary condition. Wave functions satisfy $\psi \left({\mathbf{r}}_B\right)=\psi \left[{\mathcal{R}}_{\pm \varphi }\left({\mathbf{r}}_B\right)\right]$, where ${\mathbf{r}}_B$ is a point on either boundary (with one $\mathbf{k}$ point along tube axis); operation ${\mathcal{R}}_{\pm \varphi }\left({\mathbf{r}}_j\right)$ rotates ${\mathbf{r}}_j$ an angle $\pm \varphi$ around the wedge apex. We define the vector ${\mathbf{r}}_{ij}$ from atom $i$ to atom $j$ as equal to shortest of vectors $\left[{\mathcal{R}}_{\varphi }\left({\mathbf{r}}_j\right)-{\mathbf{r}}_i\right]$, $\left[{\mathcal{R}}_{-\varphi }\left({\mathbf{r}}_j\right)-{\mathbf{r}}_i\right]$, or $\left({\mathbf{r}}_j-{\mathbf{r}}_i\right)$, where ${\mathbf{r}}_i$ and ${\mathbf{r}}_j$ are atom positions within the simulation box. For some atom pairs ${\mathbf{r}}_{ij}$ crosses the boundary and renders Newton’s third law invalid.

Figure 2

Raman spectra of bent (13,0) (6,6), and (13,4) carbon nanotubes in different polarization pictures. Polarization picture is defined by directions of incident and scattered light as shown by symbols on the right. Bending increases linearly from zero (upmost lines) to $\Theta =4.2\%$ for (13,0), $\Theta =2.6\%$ for (6,6), and $\Theta =4.8\%$ for (13,4) tubes (lowest lines; corresponds to wedge angle $\varphi =10°$ for all tubes). The symbols with arrows refer to symmetry of modes for straight tubes in line-group notation (Ref. 27). Peaks were Lorentzian broadened with full width at half maximum of $5{\text{cm}}^{-1}$. Small symbols in the spectra of (6,6) tube with $zz$ polarization refer to the subscripts of original mode symmetries.

Figure 3

(Color online) Qualitative view of the nodes and antinodes of selected vibration modes for (13,0) and (6,6) tubes. Thick blue line (dark gray) represents an antinode, red (light gray) an antinode with opposite phase and vanishing line a node; the actual direction of the vibration amplitude is either circumferential or along tube axis. Modes are for straight or bent tubes as indicated on the left. The symbols below show the polarization pictures where the bent modes become most visible. We were unable to identify one $E_3^{+}$ mode for (6,6) tube (faint symbol).

Figure 4

Variation in bond lengths for (13,0) (left) and (6,6) (right) tubes as a function of bending parameter $\Theta$. Bonds are divided into mostly parallel $(\parallel )$ and mostly perpendicular $(\perp )$ bonds with respect to tube axis (in and out refer to the sides of torus). Inset shows the effective nearest-neighbor spring constant, calculated by scaling single graphene layer; the inset was used to map the bond-length scale (left-hand axis) into spring constant scale (right-hand axis). The equilibrium spring constant 970 N/m and thermal-expansion coefficient $5.13\times {10}^{-6}{\text{K}}^{-1}$, used to make the inset, are in overall agreement with reference results (Refs. 28, 29).