Nature and strength of bonding in a crystal of semiconducting nanotubes: van der Waals density functional calculations and analytical results

The dispersive interaction between nanotubes is investigated through ab initio theory calculations and in an analytical approximation. A van der Waals density functional (vdW-DF) [M. Dion et al., Phys. Rev. Lett. 92, 246401 (2004)] is used to determine and compare the binding of a pair of nanotubes as well as in a nanotube crystal. To analyze the interaction and determine the importance of morphology, we further compare results of our ab initio calculations to a simple analytical result that we obtain for a pair of well-separated nanotubes. In contrast to traditional density functional theory calculations, the vdW-DF study predicts an intertube vdW bonding with a strength that is consistent with recent observations for the interlayer binding in graphitics. It also produces a nanotube wall-to-wall separation, which is in very good agreement with experiments. Moreover, we find that the vdW-DF result for the nanotube-crystal binding energy can be approximated by a sum of nanotube-pair interactions when these are calculated in vdW-DF. This observation suggests a framework for an efficient implementation of quantum-physical modeling of the carbon nanotube bundling in more general nanotube bundles, including nanotube yarn and rope structures.

Figure 1

(Color online) Atomic and electronic structures as well as filament organization in crystals of semiconducting (8,0) zigzag nanotubes. The lower panel shows the fully relaxed atomic configuration of the individual (8,0) nanotubes as calculated in a traditional implementation of ab initio DFT. The top left panel shows our corresponding traditional-DFT results for the length-averaged electron density concentrated at the atomic nuclei; the contour spacing is specified in steps of $0.15e{\text{Å}}^3$. Finally, the top right panel shows the hexagonal crystalline order of the nanotube bundle. We calculate the intertube dispersive interaction and determine the nanotube crystalline structure by using a recently developed ab initio van der Waals density functional approach (Ref. 22).

Figure 2

(Color online) Length and radially averaged electron density $\overline{n}\left(s\right)$ shown together with the positions of the effective and the geometric radius. The radial separation is given relative to the geometric radius of the nanotube. The background inset shows the length-averaged electron density with contour lines separated in steps of $0.15e{\text{Å}}^3$.

Figure 3

(Color online) Nanotube binding energy $E^{\text{vdW-DF}}\left(\Delta \right)-E^{\text{vdW-DF}}\left(\Delta \to \infty \right)$ (per unit length) for semiconducting (8,0) nanotubes evaluated in vdW-DF as a function of the wall-to-wall separation $\Delta$. The top panel shows vdW-DF results for the hexagonal crystal and compares the crystal interaction energy (thick solid curve) against an estimate (thin solid curve) based on a sum of vdW-DF results for the nanotube-pair interactions. The bottom panel reports the vdW-DF calculations of the binding of two parallel nanotubes (dashed curves) in three different atomic configurations indicated in the inset. The sum of those three pair interactions constitutes the approximation for the crystal interaction (thin solid curve) in the top panel.

Figure 4

Nonlocal correlation energy per unit length for a nanotube dimer near binding separations (main panel) and in the intermediate-to-asymptotic regime (inset panel). The thick dashed curves show the results of the full vdW-DF calculation of the nonlocal correlation energy (vdW interaction). The dotted and dashed-double-dotted curves show the (traditional) asymptotic interaction estimates as determined from the asymptotics $-B_5{d}^{-5}$ and $-B_5{d}^{-5}-B_7{d}^{-7}$, respectively. Finally, the dashed-dotted curve shows the analytical evaluation that approximates the electrodynamical response by the long-wavelength form but retains a full description of the nanotube morphology. The analytical result also reflects the surface-physics insight that the electrodynamical response is dominated by contributions outside the radius defined by the atomic positions (Fig. 2).