Inhomogeneity of ${}^{13}\mathrm{C}$ isotope distribution in isotope engineered carbon nanotubes: Experiment and theory

Vibrational modes of ${}^{13}\mathrm{C}$ isotope enriched single-walled carbon nanotubes are inhomogeneously broadened due to the random distribution of isotopes. We study this effect on the radial breathing mode theoretically using density-functional theory within the local-density approximation and compare the result with experiments on inner tubes in double-walled carbon nanotubes grown from ${}^{13}\mathrm{C}$-enriched fullerenes. Increased inhomogeneity was achieved by growing inner tubes from a mixture of enriched and natural fullerenes, which is explained by the calculations. This shows the absence of carbon diffusion along the tube axis during inner tube growth, supporting the theory of inner tube growth by Stone-Wales transformations from interconnected fullerenes.

Figure 1

Raman intensities (normalized to the mode intensities of ${}^{12}\mathrm{C}_{60}$) of the pinch and radial breathing modes of ${\mathrm{C}}_{60}$ versus the $w^n$ scalar product square of the distorted vibrational modes with the corresponding original modes for ${}^{13}\mathrm{C}{}^{12}\mathrm{C}_{59}$ (circles) and a randomly chosen ${}^{13}\mathrm{C}_6{}^{12}\mathrm{C}_{54}$ (diamonds). The points fit well to the $x=y$ line (dashed line), showing that $w^n$ is a good approximation of the relative Raman intensities. Note that the RBM is already “split” for ${}^{13}\mathrm{C}_1{}^{12}\mathrm{C}_{59}$ due to the mixing of the RBM with other close energy vibrational modes. This effect is much smaller for the pinch mode, which is more separated from other modes in energy than the RBM.

Figure 2

Measured Raman spectra (solid lines) of DWCNTs with ${}^{\mathrm{nat}}\mathrm{C}$, ${}^{13}\mathrm{C}_{0.15-M}$, ${}^{13}\mathrm{C}_{0.28}$, and ${}^{13}\mathrm{C}_{0.82}$ enriched inner tubes at $\lambda =676\mathrm{nm}$ laser excitation and $90\mathrm{K}$ measured with high resolution. Vertical dashed lines are intended to guide the eye. The decomposition of the 15 separate RBM lines in the ${}^{\mathrm{nat}}\mathrm{C}$ spectrum is also plotted below the measured spectrum with dashed lines.

Figure 3

(Color online) Calculated inhomogeneous broadening of the RBM mode of the (10,0), (7,4), and (5,5) SWCNTs (diamonds, squares, and crosses, respectively), compared with measurements (circles). All values are full width at half maximum. The results in the lower part of the figure show the case of uniform filling derived DWCNTs, while the few data points in the upper left corner are the case of mixed filling derived tubes. The error bars on the data points of (10,0) depict the variance of the broadening as a function of supercell size. The inset shows exactly the same data except that the calculated results are upscaled by a factor of 1.65 (see text).