Raman spectra of unfilled and filled carbon nanotubes: Theory

The Raman spectra of two $G$ bands and a radial breathing mode (RBM) of unfilled and filled single-wall semiconducting and metallic carbon nanotubes have been investigated theoretically in the presence of electron-phonon and phonon-phonon interactions. Excitation of low frequency optical plasmons in the metallic nanotube is responsible for the peak known as the Breit-Wigner-Fano (BWF) line shape in the $G$-band Raman spectra. In a filled nanotube, there is an additional peak due to excitation of the phonon of the filling atom or molecule. Positions, shapes, and relative strengths of these Raman peaks depend on the phonon frequencies of the nanotube and that of the filling atoms, and strengths and forms of the plasmon-phonon and phonon-phonon interactions. For example, filling atoms with phonon frequency close to the RBM frequency of the nanotube may broaden and lower the RBM Raman peak to such an extent that it may become barely visible. Hybridization between the $G$ bands and the filling atom phonon is also strong when these two frequencies are close to each other, and it has important effects on the $G$ band and the BWF line shapes. When the phonon frequency of the filling atom is far from the RBM and $G$-band frequencies, it gives rise to a separate peak with modest effects on the RBM and $G$-band spectra. The Raman spectra of semiconducting unfilled and filled nanotubes have similar behaviors as those of metallic nanotubes, except that normally they have Lorentzian line shapes and do not show a BWF line shape. However, if a semiconducting nanotube is filled with donor atoms, it is predicted that the BWF-type line shape may be observed near the RBM, or the $G$ band, or the filling atom Raman peak.

Figure 1

Schematic representation of a filled single-wall carbon nanotube Ref. 39. It is assumed that each filling atom is harmonically coupled to the carbon atoms of the nanotube.

Figure 2

(Color online) (a) The RBM and two $G$-band Raman spectra of an unfilled semiconducting nanotube for three values of nanotube radius and hence three values of lower $G$-band and RBM frequencies. (b) Raman spectra of unfilled semiconducting nanotube near the $G$-band frequencies. Note that all three bands are Lorentzian in form. In Fig. 2 and in all the subsequent figures, the Raman intensity $\left(I\right)$ has been expressed in arbitrary units and the normalized frequency $\omega$ has been plotted in dimensionless units.

Figure 3

(Color online) (a) The RBM and two $G$-band Raman spectra of an unfilled metallic carbon nanotube for three values of the radius of the nanotube and the corresponding three RBM and three lower $G$-band frequencies. (b) The metallic nanotubes clearly show the asymmetric and wider BWF satellite band close to the upper $G$ band.

Figure 4

(Color online) (a) Raman spectra of a filled metallic carbon nanotube with three values of the filling atom phonon frequencies close to the RBM frequency of 0.09. There is an extra peak due to the filling atom phonon, which hybridizes with the peak of the RBM phonon. (b) Raman spectra near the RBM mode. (c) Raman spectra near the $G$ bands. Note that the $G$ bands are not affected appreciably by the low frequency filling atom.

Figure 5

(Color online) (a) Raman spectra of a filled metallic carbon nanotube for three different values of $K^2$, with the filling atom frequency close to the RBM frequency. (b) Raman spectra near the RBM mode. Note the strong effect of hybridization.

Figure 6

(Color online) (a) Raman spectra of a filled metallic carbon nanotube with three values of the filling atom frequencies lying between the RBM frequency and the $G$-band frequencies. (b) Raman spectra close to the $G$ bands. (c) Raman spectra close to the RBM band.

Figure 7

(Color online) Raman spectra of a semiconducting nanotube filled with donor atoms. Three different donor electron plasmon frequencies have been chosen close to the RBM frequency. BWF-like satellite peaks appear close to RBM Raman peak due to excitation of plasmons of the donor electrons

Figure 8

(Color online) Raman spectra of a semiconducting nanotube filled with donor atoms. Three different donor electron plasmon frequencies have been chosen to be close to the filling atom phonon frequency $({\tilde{\omega }}_0=0.6)$. Note the BWF-like peaks near the filling atom phonon peak due to excitation of plasmons of donor electrons.