Raman Active Phonons of Identified Semiconducting Single-Walled Carbon Nanotubes

The resonant Raman spectra of $(n,m)$ semiconducting single-walled carbon nanotubes, unambiguously identified from their electron diffraction patterns, have been measured. The diameter dependence of the frequency of the tangential modes with $A$ symmetry has been obtained in the diameter range from 1.4 to 2.5 nm. The comparison between the excitation energies and the calculated transition energies allowed us to determine precisely the values of the $E^s{}_{33}$ and $E^s{}_{44}$ transition energies. Finally, in the debate concerning the dominant process at the origin of the first-order Raman scattering in single-walled carbon nanotubes (single resonance process or double resonance process), our results are well understood in the framework of a single resonance process.

Figure 1

(a) Radial breathing modes and (b) tangential modes with $A$ symmetry of the $(11,10)$ SWCNT, $E_{\mathrm{Laser}}=2.41\mathrm{eV}$ (top), $(17,9)$ SWCNT, $E_{\mathrm{Laser}}=2.41\mathrm{eV}$ (middle), and $(27,4)$ SWCNT, $E_{\mathrm{Laser}}=1.92\mathrm{eV}$ (bottom). Experimental data: black symbols. Each component of a RBM and TM bunch is fitted by a Lorentzian profile (thin blue solid line). Calculated total profile: thick (red) solid line.

Figure 2

Dependence of the ${\mathrm{TM}}^{+}$ (black symbols) and ${\mathrm{TM}}^{-}$ (red symbols) frequencies as a function of the diameter. Solid squares: measurements on $(n,m)$ identified SWCNTs. Open circles: measurements on tubes for which the precise diameters were determined from the recording of the equatorial line of the electron diffraction pattern. Open triangles: data obtained on freestanding nonidentified SWCNTs. The diameters of these latter tubes were obtained from the frequency of their RBM by using the RBM frequency vs diameter relationship (see text). Solid (red) line: ${\mathrm{TM}}^{-}$ vs diameter, from Ref. 12. The dashed (black) line indicates the average value of all ${\mathrm{TM}}^{+}$ frequencies.

Figure 3

(a) Radial breathing modes and (b) tangential modes with $A$ symmetry of the $(15,14)$ SWCNT. $E_{\mathrm{Laser}}=1.92\mathrm{eV}$ (top), $E_{\mathrm{Laser}}=2.41\mathrm{eV}$ (bottom). Experimental data: black symbols. Each component of a RBM and TM bunch is fitted by a Lorentzian profile (thin blue solid line). Calculated total profile: thick (red) solid line.

Figure 4

Optical transition energies vs diameter. Experimental results (open and solid red dots). Nonorthogonal tight-binding calculations: semiconducting I ($[(2n+m)\mathrm{mod}3=1]$ (black solid squares) and semiconducting II ($[(2n+m)\mathrm{mod}3=2]$ (black open squares). The large (blue) triangles indicate the calculated transition energies for the $(n,m)$ SWCNTs investigated by Raman spectroscopy. The calculated energies have been rigidly upshifted by 0.45 eV (see text).