Modification of the electronic structure of DWNTs due to interlayer interactions

Modification of the electronic structure of DWNTs due to interlayer interactions

Fig. 1. Schematic of the atomic structure of a 100 A-long piece the DWNT (15,13)@(21,17). The radius of the inner (outer) layer is 9.51 A (12.91 A) and the interlayer separation is 3.40 A. A clear moire pattern can be observed.

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 [1,2].

However, even though the separate layers of a DWNT have translational symmetry, their translational periods are generally incommensurate and the DWNT has no translational symmetry, which poses a major problem before the calculation of various properties of the DWNTs.

It is important to address this problem because the interlayer interactions modify the electronic structure of the DWNT, resulting in measurable effects on the optical transitions, which can be shifted by tens of meV.

The straightforward approach to solving the electronic structure problem for the DWNTs relies on the use of first-order quantum-mechanical perturbation theory to derive the interlayer interactions induced shift of the optical transitions. This, however, implies that such shifts are the only result of the interlayer interactions, which is not necessarily so.

Here, we follow a different route, based on the use of the recursion method within a non-orthogonal tight-binding model with four orbitals per carbon atom to directly derive the electronic density of states (DOS) and optical absorption of the DWNTs [3]. This approach allows to predict not only shifts of the optical transitions but also appearance of new optical transitions in agreement with recent experimental data.

The adopted approach is illustrated in the case of DWNT (15,13)@(21,17) (Fig. 1). A long piece of the DWNT of length 2000 A and number of orbitals 430000, is found sufficient for achieving convergence of the DOS. Similarly, a smaller piece of the DWNT of length 400 A and number of orbitals 86000 is found sufficient for convergence of the optical absorption.

Figure 2 shows the DOS of the DWNT in comparison with the DOS of the non-interacting layers. It is clear that the DOS of the DWNT undergoes significant changes, most of the peaks being red- or blue-shifted with respect to those of the non-interacting layers. Since the number of spikes of the DOS of the DWNT corresponds to that of the non-interacting layers, it is tempting to derive the optical transitions of the DWNT as the separation between the mirror spikes of the DOS and adopt the same notation as for the non-interacting layers. However, the analysis bellow shows that this can lead to incorrect conclusions.

Figure 3 shows the DOS of the DWNT in comparison with the contributions of the separate interacting layers. It is clear that most of the spikes of the DOS of the DWNT can be connected to one of the layers. However, four spikes, marked by vertical lines and connected by arrows 1 to 4, have non-negligible contribution from both layers, which can be interpreted as a significant mixing of the electronic states of the two layers. Thus, four transitions can be expected to appear because of the mixing of the states: two transitions 1 and 4 between mirror spikes, corresponding to transitions Sⁱ₄₄ and S^o₅₅ of the non-interacting layers, as well as two cross-band transitions 2 and 3. Similarly, large mixing of the electronic states is evident for the spikes, connected by arrows 1′ to 4′. Therefore, four transitions can be expected between these spikes: two transitions 1′ and 4′ between mirror spikes, corresponding to transitions Sⁱ₂₂ and S^o₃₃ of the non-interacting layers, aa well as two cross-band transitions 2′ and 3′. The evidenced mixing of the states of the layers alone does not allow to draw conclusions on the observability of the mentioned transitions, as the height of the absorption peaks will also depend on the interaction between the electronic states of the layers.

Figure 4 shows the calculated optical absorption of the DWNT (black line) in comparison with that of the non-interacting layers (red line). It is clearly seen that the absorption spectrum of the DWNT differs significantly from that of the non-interacting layers in shape and number of peaks. As suggested above, four peaks (1 to 4) can be expected instead of the two peaks Sⁱ₄₄ and S^o₅₅ of the non-interacting layers. It is seen that at the energies of transitions 1 and 4 there is only a tiny kink and a small bump, while at the energies of the cross-band transitions 2 and 3 there is a high peak. Therefore, the cross-band transitions give major contribution to the absorption, while the mirror transitions give insignificant contributions. Similarly, transitions 1′ and 4′ give rise to smaller peaks that for the non-interacting layers, while the cross-band transitions 2′ and 3′ contribute to a wide peak.

The obtained behavior of the absorption peaks can be explained by simple arguments, provided by the quantum-mechanical perturbation theory [3]. Simple expressions, allowing to evaluate the height of the peaks of the DWNT for a given difference of the optical transitions of the inner and outer layer, and interaction matrix element, are derived [3].

In conclusion, using the recursion method within a non-orthogonal tight-binding model, we show that the interaction between the layers of a DWNT can give rise to major changes of the electronic structure of the DWNT, such as mixing of the electronic states of the two layers, shift of the optical transitions as well as appearance of new transitions, which is evidenced here for the first time. Strong mixing of the states can be expected if the two layers have close optical transitions and there is a strong interaction between the states of the two layers (e.g., Sⁱ₄₄ and S^o₅₅). Significant mixing of the electronic states can be expected even for relatively large difference of the transition energies of the layers (e.g., Sⁱ₂₂ and S^o₃₃), which can result in absorption peaks of the cross-band transitions, comparable to those of the mirror transitions.

We underline that the first-order quantum-mechanical perturbation theory commonly used for assessing the shift of the electronic transitions in DWNTs cannot grasp sufficiently well the complex interaction-induced changes of the electronic structure of DWNTs and does not predict the additional cross band transitions, which have been observed recently.

References:

1. V. N. Popov, New J. Phys. 6 (2004) 17.

2. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407/1-12.

3. V. N. Popov, Carbon 170 (2020) 30-36.

Fig. 2. The DOS of the DWNT (15,13)@(21,17) (black line) in comparison with the DOS of the non-interacting inner layer (15,13) (blue line) and outer layer (21,17) (red line). The horizontal arrows between the mirror spikes mark the optical transitions for the non-interacting layers, denoted by Sⁱ and S^o for the inner and outer layer, respectively.

Fig. 3. The DOS of the DWNT (15,13)@(21,17) (black line) in comparison with the contributions of the inner layer (15,13) (blue line) and outer layer (21,17) (red line). The graph shows strong mixing of electronic states of the two layers close to energies, marked by vertical lines. The red and blue horizontal arrows show transitions between mirror spikes corresponding to specific pairs of transitions of the layers, while the black horizontal arrows show cross-band transitions, induced by the mixing of the electronic states.

Fig. 4. The absorption coefficient of the DWNT (15,13)@(21,17) (black line) in comparison with that for a DWNT without interlayer interaction (red line). The red vertical lines mark the transitions of the layers. The red vertical lines mark the transitions of the non-interacting inner and outer layers. The black vertical lines mark the transitions of the DWNT. The numbers 1 to 4 and 1′ to 4′ mark the transitions, shown in Fig. 3.

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Valentin Popov

September 20, 2020