Raman spectra of single-walled carbon nanotubes with vacancies

Density-functional based tight-binding method in conjunction with nonresonant bond-polarization theory is used to predict Raman spectra of single-walled carbon nanotubes with vacancies. Calculations show that the high-energy Raman active modes are more sensitive to vacancies and their energy is generally lowered, while also the radial breathing mode may change in intensity and shape. The connection of these observations to experiments is discussed. The effect of vacancies on the spatial nature of Raman active eigenmodes is analyzed and discussed.

Figure 1

(Color online) High-energy Raman active modes of the zigzag tubes from (7,0) to (21,0). Different modes are separated by different symbols and different types of tubes with colors, with green denoting mod1, blue mod2 semiconductors, and red mod0 metallic tubes.

Figure 2

Energy gaps for zigzag tubes from (7,0) to (21,0) (filled squares) compared to reference tight-binding results of Ref. 23 (empty squares). mod1 and mod2 semiconductive tubes are separated with lines.

Figure 3

(Color online) The energy gap as a function of the maximum displacement along the three longitudinal Raman active modes of metallic (12,0) tube.

Figure 4

(a) All Raman active modes of armchair tubes from (5,5) to (12,12) tube (filled squares) with reference results from Ref. 34 (empty squares). (b) $\omega =a$/diameter fit to four lowest Raman active modes of zigzag from (7,0) to (21,0) and armchair tubes from (5,5) to (12,12) and also to the corresponding modes in six chiral tubes (6,4), (8,4), (9,3), (11,2), and (13,4). Fitting parameter $a$ for the RBM is shown in the figure.

Figure 5

Raman active spectra for uniform periodical (13,0) tube and (13,0) tubes with different vacancy systems. Dotted line denotes $zz$ $\left(A_0^{+}\right)$, dashed line $xz$ $\left(E_1^{-}\right)$, and solid line $xy$ $\left(E_2^{+}\right)$ incident and/or scattered light polarizations (symmetries). The spectra are the Lorentz broadened with full width at half maximum $\left(\mathrm{FWHM}\right)=5{\mathrm{cm}}^{-1}$, which produces observation limit close to experimental one.

Figure 6

(Color online) (a) The lowest Raman active mode $E_2^{+}$, (b) radial breathing mode, and (c) the highest $A_0^{+}$ mode in uniform tube (left-hand side panels) and in defective tubes (right-hand side panels). The defective tubes are the ones with two vacancies at opposite sides of the tube [in (a) and (b)] and a tube with one vacancy [in (c)].

Figure 7

Raman active modes of (13,0) tube with one vacancy, concentration of $1∕208$ atoms, with the bond-polarization method (dotted line) and the introduced eigenvector overlap calculation according to Eq. (9) (solid line). Three light polarization spectra $xy$, $xz$, and $zz$ are separately drawn and the peaks are the Lorentz broadened by $\mathrm{FWHM}=5{\mathrm{cm}}^{-1}$. In the overlap calculation, the modes in the defective tube are compared to the eigenvectors and their Raman intensities (calculated by using bond–polarization method) of the uniform (13,0) tube.

Figure 8

Averaged magnitude of the vibrations inside different distances from vacancy for the highest $E_1^{-}$ symmetry corresponding modes in Fig. 5b type of one vacancy system (empty symbols). Percentages denote maximum overlap for both represented vacancy modes with respect to original degenerate $E_1^{-}$ mode in uniform tube (filled symbols). Dotted line shows the radius inside of which the vacancy considerably affects the motion of the atoms.

Figure 9

Electronic density of states near Fermi energy for semiconducting (7,0) uniform tube below and the same tube with average single vacancy consentration of $1∕336$ atoms above. The Fermi surface in the figure is at $-5.2\mathrm{eV}$. The solid and dashed curves denote occupied and unoccupied states, respectively.