Isotope effect on phonon spectra in single-walled carbon nanotubes

Carbon nanotubes are usually made up of ${}^{12}\mathrm{C}$ atoms. By adopting the force constant model, the effects of isotope doping with ${}^{13}\mathrm{C}$ on the vibrational, optical, and thermal properties of single-walled carbon nanotubes are theoretically investigated, including the phonon eigenmodes and density of states, the Raman scattering spectra, and specific heat. Three types of doping patterns (axial stripes, circumferential bands of varying width, and random) are studied, in which an emphasis is put on the isotope-nanotube-junctions. Several new effects of the isotope doping in nanotubes have been found, including the confined optical phonon modes due to the spatial separation of isotopes, the red-shift of Raman peaks in general and blue-shift anomaly of the confined LO peaks in specific tube chirality and doping configuration, and the enhancement of the specific heat due to the ${}^{13}\mathrm{C}$-doping. This work provides with a theoretical guide for isotope labeling, which is important for understanding the growth mechanism of the carbon nanotubes.

Figure 1

Illustration of doping Configuration B in armchair (a) and zigzag (b) SWNT’s.

Figure 2

Isotope effect on phonon dispersion relations in (10,10) SWNT’s with a doping percentage of 50%. (a) Pure ${}^{12}\mathrm{C}$SWNT’s, (b) Configuration A1,(c) Configuration A2, and (d) Configuration A3.

Figure 3

Isotope effect on phonon density of states in a (10,10) SWNT with a doping percentage of 50% (Configuration A).

Figure 4

Isotope effect on phonon dispersion relations in a (10,10) SWNT (Configuration B). (a) Undoped ${}^{12}\mathrm{C}$SWNT’s; (b) ${\left({}^{12}\mathrm{C}\right)}_1\text{−}{\left({}^{13}\mathrm{C}\right)}_1$ junction; (c) ${\left({}^{12}\mathrm{C}\right)}_2\text{−}{\left({}^{13}\mathrm{C}\right)}_2$ junction; and (d) ${\left({}^{12}\mathrm{C}\right)}_4\text{−}{\left({}^{13}\mathrm{C}\right)}_4$ junction.

Figure 5

Isotope effect on phonon density of states in a (10,10) SWNT (Configuration B).

Figure 6

Linear-chain model in (10,10) SWNT’s doped in Configuration B.

Figure 7

Calculated Raman intensity of a (10,10) SWNT in the $z\left(xx\right)\overline{z}$ configuration.

Figure 8

Isotope effect on Raman scattering spectra in a (10,10) SWNT. (a) Configuration A1, (b) Configuration A2, (c) Configuration B, ${\left({}^{12}\mathrm{C}\right)}_8\text{−}{\left({}^{13}\mathrm{C}\right)}_8$ junction, (d) Configuration B, ${\left({}^{12}\mathrm{C}\right)}_{16}\text{−}{\left({}^{13}\mathrm{C}\right)}_{16}$ junction.

Figure 9

Comparison between theory and experiment of the specific heat in (10,10) SWNT’s.

Figure 10

Coefficient $a\left(T\right)$ in $(n,n)$ SWNT’s.

Figure 11

Coefficient $a\left(T\right)$ in $(n,0)$ SWNT’s.