Pressure-induced Raman-active radial breathing mode transition in single-wall carbon nanotubes

Using classical constant-pressure molecular dynamics simulations and the force constants model, radial breathing mode (RBM) transition of single-wall carbon nanotubes under hydrostatic pressure is reported. With the pressure increased, the RBM shifts linearly toward higher frequency, and the RBM transition occurs at the same critical pressure as the structural transition. The group theory indicates that the RBMs are all Raman-active; however, due to the effect of the frequency transition and the electronic structure change for tube radial deformation, the Raman intensity of the modes becomes so weak as not to be experimentally detected, which is in agreement with a recent experiment by S. Lebedkin et al. [Phys. Rev. B 73, 094109 (2006)]. Furthermore, the calculated RBM transition pressure is well fitted to the cube of diameter $(\sim 1∕d^3)$.

Figure 1

Calculated volume [scaled by the volume at zero pressure $V_0=1772{\mathrm{\AA }}^3$ for (10,10) tube and $V_0=1027{\mathrm{\AA }}^3$ for (10,0) tube, respectively] and energy as a function of pressure for (10,10) (line with quadrangle), (10,0) (line with circle), and (4,8) (line with triangle) SWNTs at $300\mathrm{K}$ with transition at 1.06, 5.50, and $4.70\mathrm{GPa}$, respectively.

Figure 2

The calculated RBM frequency vs pressure for (10,10) and (10,0) SWNTs at $300\mathrm{K}$. A pressure-induced RBM frequency transition occurs at 1.06 and $5.50\mathrm{GPa}$, exactly where the structure transition occurs.

Figure 3

The atomic displacement patterns of RBMs for (10,10) SWNT at selected pressures of 0.90, 1.06, 1.10, 1.15, 1.20, and $1.25\mathrm{GPa}$. The calculated atomic displacements of the atoms are depicted by arrows. $a_1$, $a_2$, and $b_1$, ${\mathrm{b}}_2$ are marked in the short and long axes of the oval shape at $1.20\mathrm{GPa}$, respectively.

Figure 4

The calculated amplitudes of unit cell atoms arranged in a line for (10,10) SWNT at selected pressures of 0.90, 1.06, 1.10, 1.20, and $1.25\mathrm{GPa}$. Dotted lines sign the atomic positions with amplitudes changing obviously as $a_1$, $a_2$, and $b_1$, $b_2$, which correspond to the short and long axes of an elliptical shape in Fig. 3 (see $1.20\mathrm{GPa}$).

Figure 5

The calculated RBM transition pressure vs the diameter for $(n,n)$, $(n,0)$, $(n,2n)$ SWNTs at $300\mathrm{K}$. The transition pressure can be well fitted to $\sim 1∕d^3$ for the line with triangle, circle, and quadrangle. The inset pattern shows that the transition pressure is the direct proportion of the inverse of $d^3$.