Linear augmented cylindrical wave method for calculating the electronic structure of double-wall carbon nanotubes

Abstract

Electronic structure of double-wall carbon nanotubes (DWNTs) consisting of two concentric graphene cylinders with extremely strong covalent bonding of atoms within the individual graphitic sheets, but very weak van der Waals type interaction between them is calculated in the terms of the linear augmented cylindrical wave (LACW) method. A one-electron potential is used and the approximations are made in the sense of muffin-tin (MT) potentials and local density functional theory only. The atoms of DWNT are considered to be enclosed between cylinder-shaped potential barriers. In this approach, the electronic spectrum of the DWNTs is governed by the free movement of electron in the interatomic space of two cylindrical layers, by electron scattering on the MT spheres, and by electron tunneling between the layers. We have calculated the complete band structures and densities of states in the Fermi level region of the purely semiconducting zigzag DWNTs$(n,0)@({n}^{\ensuremath{'}},0)$ ($10\ensuremath{\leqslant}n\ensuremath{\leqslant}23$ and $19\ensuremath{\leqslant}{n}^{\ensuremath{'}}\ensuremath{\leqslant}32$) with interlayer distance $3.2\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}\ensuremath{\leqslant}\ensuremath{\Delta}d\ensuremath{\leqslant}3.7\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$. Analogously data are obtained for metallic armchair $(n,n)@({n}^{\ensuremath{'}},{n}^{\ensuremath{'}})$ nanotubes ($n=5$ or 4 and ${n}^{\ensuremath{'}}=10$ or 9). According to the LACW calculations, the interwall coupling results in a distinctly stronger perturbation of the band structure of inner tube as compared to that of the outer one. In the case of semiconducting DWNTs, the minimum gap ${E}_{11}$ between the singularities of the conduction and valence bands of the shell tubules decreases from $0.15\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}0.22\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ or increases from $0.7\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}0.15\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, if dividing ${n}^{\ensuremath{'}}$ by three leaves a remainder of 1 or 2, respectively. In both cases, the $\ensuremath{\Delta}{E}_{11}$ shifts of the gap do not decay, but slightly oscillate as one goes to the tubules with larger diameters $d$. For inner tubules, the $\ensuremath{\Delta}{E}_{11}$ shift depends strongly on the $d$. For $n\phantom{\rule{0.2em}{0ex}}\mathrm{mod}\phantom{\rule{0.2em}{0ex}}3=2$ series with $10\ensuremath{\leqslant}n\ensuremath{\leqslant}16$, the shifts $\ensuremath{\Delta}{E}_{11}$ are positive, the maximum values of $\ensuremath{\Delta}{E}_{11}$ being equal to 0.39 and $0.32\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, respectively. As one goes to the inner tubules with larger diameters, the shift $\ensuremath{\Delta}{E}_{11}$ quickly decays and thereupon varies between 0.06 and $\ensuremath{-}0.05\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. In the case of armchair DWNTs, the interlayer coupling does not break down the metal-type character of the band structure of the tubules. The high-energy shift of the $\ensuremath{\sigma}$ states relative to the occupied $\ensuremath{\pi}$ states is seen to be the most significant effect of the interlayer interaction in the armchair double-wall pair. The large shifts of optical gaps of the tubules due to formation of the DWNTs complicate the determination of the structure of DWNTs on the basis of optical data. On the other hand, the results obtained open the opportunity to classify experimental data on the DWNTs more specifically.