Calculation of Raman-active modes in linear and zigzag phases of fullerene peapods

We report on minimum energy calculations, using a convenient Lennard-Jones expression of the van der Waals intermolecular potential, to derive the optimum configurations of ${\mathrm{C}}_{60}$ molecules inside single wall carbon nanotubes. Depending on the diameter of the nanotube, ${\mathrm{C}}_{60}$ molecules were found to form linear or zigzag chains inside the nanotubes. In the following, we use the spectral moments method, together with a bond-polarizability model, to calculate the nonresonant Raman spectrum for infinitely long isolated ${\mathrm{C}}_{60}$ peapods. We present the evolution of the Raman spectrum as a function of the diameter and chirality of the nanotube. The changes of the Raman spectrum as a function of the configuration of the ${\mathrm{C}}_{60}$ molecules inside the nanotubes are identified. On the other hand, the effect of the filling factor on the Raman spectrum is analyzed. These predictions are useful to interpret the experimental Raman spectra of fullerene peapods.

Figure 1

Schematical representation of carbon peapods showing some of the parameters used for the geometrical optimization of the ${\mathrm{C}}_{60}$ molecules inside the nanotube (see text). The letter $C$ indicates the position of the center of one of the ${\mathrm{C}}_{60}$ molecule.

Figure 2

The $VV$ (top) and $VH$ (bottom) calculated Raman spectra in the BLM (left) and TLM (right) ranges of peapods: ${\mathrm{C}}_{60}@(10,10)$ (curve $c$) and ${\mathrm{C}}_{60}@(13,13)$ (curve $b$). The calculated Raman spectra of the empty SWCNTs are displayed: (10,10) (curve $a$, dashed line) and (13,13) (curve $a$, solid line).

Figure 3

The diameter dependence of the RBLM frequency in linear and zigzag peapods: high-frequency component, RBLM1, (star) and low-frequency component, RBLM2, (cross). The diameter dependence of the RBM for the related empty tubes is also displayed (open circles). The limit ($D_c$, see text) between linear and zigzag peapods is identified by the vertical dashed line. Inset: the corresponding intensities of the RBLM1 and RBLM2 versus the diameter of the nanotubes.

Figure 4

Chirality dependence of the $VV(\equiv ZZ)$ polarized Raman spectrum. Linear peapods (top): ${\mathrm{C}}_{60}@(17,0)$ (curve $a$), ${\mathrm{C}}_{60}@(14,5)$ (curve $b$), ${\mathrm{C}}_{60}@(10,10)$ (curve $c$). Zigzag peapods (bottom): ${\mathrm{C}}_{60}@(22,0)$ (curve $d$), ${\mathrm{C}}_{60}@(22,1)$ (curve $e$), and ${\mathrm{C}}_{60}@(13,13)$ (curve $f$). Spectra are displayed in the BLM (left-hand side) and TLM (right-hand side) regions.

Figure 5

The $VV(\equiv ZZ)$ polarized Raman spectra displayed in the BLM region of infinite peapods as a function of the filling factor. The filling factor is 20%, 40%, 60%, 80%, and 100% from bottom to top. Linear peapods: ${\mathrm{C}}_{60}@(10,10)$ (curve $a$) and ${\mathrm{C}}_{60}@(18,0)$ (curve $c$). Zigzag peapods: ${\mathrm{C}}_{60}@(11,11)$ (curve $b$) and ${\mathrm{C}}_{60}@(13,13)$ (curve $d$).