Shift of the G mode of double-walled carbon nanotubes due to interlayer interactions

Fig. 1. Schematic of the atomic structure of a double-walled carbon nanotube.	The double-walled carbon nanotube (DWNT) can be viewed as consisting of two coaxial cylindrical graphitic layers (Fig. 1). The separate layers have generally very large translational unit cells. The calculation of the electronic structure and phonon dispersion of the separate layer is facilitated by the use of the chiral symmetry of the layers, which allows to use a two-atom unit cell. However, even though the separate layers of a DWNT have translational symmetry, their translational periods are generally incommensurate and the DWNT as a whole has no translational symmetry. This fact poses a major problem before the calculation of various properties of the DWNTs. In particular, the weak interlayer interactions modify the electronic structure and phonon dispersion of the separate layers resulting in measurable effects: (1) the optical transitions of the separate layers are shifted by a few tens of meV and (2) the Raman-active phonons of the separate layers are shifted by up to a few tens of cm^-1. Here, we are focusing on the shift of the two components of the Raman G band of DWNTs. The straightforward approach to solving the electronic structure and phonon dispersion problems relies on the quantum-mechanical perturbation theory, but requires substantial computational efforts. Here, we follow a different, though less rigorous, route [3]. Namely, the shift of the G band frequency is derived as two contributions: -the first contribution is the shift of the G band of the separate layers, relaxed for the DWNT using a tight-binding model for the separate layers and a Lennard – Jones potential for the interlayer interactions; -the second contribution is the shift of the G band comes from the interlayer interactions of the layers, relaxed for the DWNT. This contribution is considered slightly dependent on the layer curvature and is taken over from calculations for bilayer graphene. The relaxation of the DWNT structure was done for all inner layers with radii from 3.7 to 11.2 Å and all outer layers, for which the interlayer separation D falls in the interval from 3 to 4 Å. We find that for D < 3.4 Å, the layers experience “positive” pressure due to the interlayer interactions, resulting in shrinking of the inner layer and expansion of the outer layer. For D > 3.4 Å, the layers experience “negative” pressure due to the interlayer interactions, resulting in expansion of the inner layer and shrinking of the outer layer [3]. The relative shrinkage/expansion of the inner (i) and outer (o) layers are found to be approximately equal in absolute value: Experimentally, the shift of the G band is measured by Raman spectroscopy on DWNTs by comparison to the measures G band frequencies of the isolated layers (single-walled carbon nanotubes). However, the DWNT layers’ radii are measured insufficiently accurately, while the layers’ indices can be determined accurately. Figures 2 and 4 show the calculated G band shifts for the lower and higher G band components, G^− and G^+, as a function of the radius Riu of the inner layer for the rolled-up layers with carbon-carbon separation of 1.42 Å for several interlayer separations Du = Rou – Riu. It is seen that both G band frequencies are increased for D < 3.4 Å and decreased for D > 3.4 Å. Figures 3 and 5 show the calculated shifts of the G bands in comparison with available experimental data [3]. It can be seen that the calculations agree well with experiment within the experimental error bars. It can be concluded that the adopted approximation scheme reasonably well describes the G band shifts and can be used in the analysis of the Raman spectra of DWNTs. References: 1. V. N. Popov, New J. Phys. 6 (2004) 17. 2. V. N. Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 085407. 3. V. N. Popov, D. I. Levshov, J.-L. Sauvajol, and M. Paillet, Phys. Rev. B 97 (2018) 165417.
Fig. 2. Calculated Δ(G^−) vs the inner layer radius Riu of the rolled-up layer for several interlayer separations Du.
Fig. 3. Calculated Δ(G^−) vs the interlayer separations Du in comparison with experimental data.
Fig. 4. Calculated Δ(G^+) vs the inner layer radius Riu of the rolled-up layer for several interlayer separations Du.
Fig. 5 Calculated Δ(G^+) vs the inner layer radius Riu of the rolled-up layer for several interlayer separations Du

Back

Valentin Popov

January 28, 2018