Band structure and inter-tube optical transitions in double-walled carbon nanotubes

Usually, in optical spectra of double-walled carbon nanotubes (DWCNTs), weak van der Waals coupling between the layers leads only to a small shift of transition energies with respect to their values in pristine single-walled nanotubes. However, recent results have shown that the Rayleigh spectrum of the DWCNT (12,11)@(17,16) contains additional peaks. Using the tight-binding approximation, we demonstrate that in specific DWCNTs the interlayer coupling can slightly modify the band structure of pristine nanotubes in such a way that the unconventional inter-tube electronic transitions become possible and additional peaks in the DWCNT optical spectrum appear. Using the known experimental data on 118 optical transitions in DWCNTs, in addition to the recently published case, we reveal six more DWCNTs with inter-tube transitions and obtain geometrical selection rules permitting them. In a few dozen of DWCNTs our approach yields the energies of electronic transitions close to the experimentally observed ones and may be useful for structural identification of this type of nanotubes.

Figure 1

The reciprocal space of the DWCNT (16,9)@(24,10). (a) and (b) The cases of the same (a) and the opposite (b) handedness of the comprising nanotubes. The extended Brillouin zones of the inner and outer SWCNTs are shown in black and red, respectively. Coincident cutting lines of both tubes are denoted with gray parallel lines. With the used definition of basis vectors in direct and reciprocal spaces of SWCNT (see Appendix pp1), the cutting lines coincide automatically. In (b), the reciprocal space of the outer tube is reflected relative to the horizontal direction compared to the one shown in (a). Large black circles show modes with strong electronic coupling and correspond to translationally equivalent states of the inner tube. In pristine tube the energies of these states are equal, but the phases $\phi$ [see Eq. (2)] at points near $K_2$ and $K_3$ differ from the phase at the point near $K_1$ by $\pm 2\pi /3$. For the outer tube, the same circles correspond to different states.

Figure 2

Reciprocal space and dispersion relations of the DWCNT (12,12)@(21,13) (a)–(c) and (10,6)@(14,13) (d)–(f). In (a) and (d), the extended Brillouin zones of the inner and outer SWCNTs are shown in black and red, respectively. The cutting lines of the DWCNTs are gray; the line associated with the inter-tube transition is highlighted with black. The projections of the points $K_1^{{}^{\prime }}$ and $K_1$ on the highlighted lines have very close $k$ coordinates, which corresponds to the proximity of extrema on the dispersion curves of pristine SWCNTs shown in (b) and (e). The interlayer coupling shifts the VHS positions and modifies the DWCNT spectra, as shown in (c) and (f). Dispersions for the inner and outer tubes are shown in black and red, respectively. The black circles in (b), (c), (e), and (f) indicate the exact VHS positions obtained within the model. Arrows denote inter-tube direct transitions in DWCNTs. In all the plots, the VHS coordinate of the outer pristine SWCNT is taken as the origin ($k=0$).

Figure 3

Inter-tube transitions in DWCNT (12,11)@(17,16). (a) The superimposed extended Brillouin zones of the inner and outer SWCNTs which are shown in black and red, respectively. The cutting lines of the DWCNT are gray; lines $\mu =-1$ and $\mu =2$ highlighted with black correspond to the DWCNT bands which dispersions are presented in (b) and (c), respectively. Dispersions for the inner and outer tubes are shown in black and red, respectively. The origin in the plots is chosen as the VHS coordinate of the inner pristine SWCNT. The VHS positions are denoted by black circles, while inter-tube transitions are denoted by arrows.