Characteristics of bamboo defects in peapod-grown double-walled carbon nanotubes

Single-walled carbon nanotubes can function as nanoscale reaction chambers for growing smaller nanotubes within the host tube from encapsulated fullerenes by annealing. The diameter of the host outer tube restricts the diameter of the inner tube due to van der Waals interactions but not its chirality: it is possible that inner tubes with different chiralities start to grow in different places at the same time. A straight junction occurs at the connection of these two tubes which we refer to as bamboo defects. We show that localized states appear in the calculated density of states associated with these bamboo defects, some of them close to the Fermi level, and present a detailed theoretical study of ballistic transport through double-walled tubes where the inner shell contains bamboo defects. We find that the presence of bamboo defects should be possible to detect through electronic-transport measurements and the number of bamboo defects per unit length can be extracted from the structure of the resonances appearing in the transmission coefficient.

Figure 1

(Color online) The geometry of the bamboo defect in a (6,4)-(4,6) nanotube junction. The top and bottom panels are different views of the same geometry. Pentagons and heptagons are highlighted within the junction. The diameter of the tube expands near the pentagon defects and contracts near the heptagon defects. There are two nonequivalent junctions in the unit cell, the length of which depends on the density of junctions ($65\text{Å}$ in this case).

Figure 2

(Color online) The helical translation parameter of the screw-axis symmetry operation and the radial coordinate of the atoms of the (6,4) nanotube in a (6,4)-(4,6) nanobamboo geometry. The origin of the $z$ axis is placed at one of the junctions, and the figure is plotted up to half way toward the next junction. The parameters converge toward the values of the pristine nanotube as we move away from the junction. The rotation parameter of the screw axis (not shown) is more-less constant and corresponds to the pristine value.

Figure 3

(Color online) LDA electronic density of states for the (6,4)-(4,6) nanobamboo geometry (solid) with $32.54\text{Å}$ between neighboring junctions compared with the density of states of the pristine (6,4) nanotube (dashed).

Figure 4

(Color online) LDA electronic density of states for the (6,4)-(4,6) nanobamboo geometry (solid) with $6.26\text{Å}$ between neighboring junctions compared with the density of states of the pristine (6,4) nanotube (dashed).

Figure 5

(Color online) The square of the magnitude of the LCAO coefficients of the HOMO, LUMO, HOMO-5, and $\text{LUMO}+5$ electron states in the first half of the (6,4)-(4,6) nanobamboo unit cell (i.e., the LCAO coefficients are the coefficients $c_i$ obtained from the Hückel calculations, $\widehat{H}{\underset{̱}{c}}_i=E_i\widehat{S}{\underset{̱}{c}}_i$, where $\widehat{H}$ and $\widehat{S}$ are the Hamiltonian and overlap matrices, respectively, and ${\underset{̱}{c}}_i$ is the $i\text{th}$ solution of the eigenvalue equation with eigenvalue $E_i$, with each component of the vector ${\underset{̱}{c}}_i$ corresponding to one of the atoms in the nanotube). The states closest to the Fermi level are strongly localized while the states far from the Fermi level are delocalized as expected.

Figure 6

(Color online) Transmission probability of the (6,4)@(10,10) DWCNT obtained with the IMH model within the Landauer-Büttiker formalism. The dotted (black) line shows the number of open channels through the (10,10) nanotube while the solid (red) and dashed (blue) curves correspond to the transmission of the DWCNT when the inner shell is a (6,4)-(4,6) nanobamboo and a pristine (6,4) nanotube, respectively. The inset shows the transmission through a fullerene-nanotube peapod [${\text{C}}_{60}$ molecules encapsulated in a (10,10) nanotube].

Figure 7

(Color online) Low-energy region of the transmission probability of the (6,4)@(10,10) DWCNT. The curve labeled DWCNT depicts the case when the inner shell is a pristine (6,4) while the ones labeled Bamboo3 and Bamboo9 depict the cases when the inner shell is a (6,4)-(4,6) nanobamboo geometry. The Bamboo3 geometry contains 3 junctions which are roughly $80\text{Å}$ apart, while the Bamboo9 geometry contains nine junctions which are roughly $30\text{Å}$ apart. (In both cases full periodic boundary conditions were applied to the outer shell geometry which was of length $340\text{Å}$.) The curve labeled peapod depicts the transmission through a peapod geometry. The four curves are shifted along the transmission axis for sake of clarity. For the top two curves, only the maximum transmission (2.0) is printed on the axes with the 2.0 on the left corresponding to the peapod curve and the 2.0 on the right corresponding to the DWCNT curve. For the bottom two curves, the scale on the left side corresponds to the Bamboo9 curve while the scale on the right side corresponds to the Bamboo3 curve.

Figure 8

(Color online) The effect of traditional disorder on the transmission probability of the (6,4)@(10,10) nanobamboo DWCNT. The curve labeled Bamboo9 depicts the case when the inner shell is a (6,4)-(4,6) nanobamboo geometry containing nine bamboo junctions which are roughly $30\text{Å}$ apart. The curve labeled $\text{Bamboo}9+\text{SW}$ depicts the case when nine additional Stone-Wales pair defects are added to the previously mentioned geometry, randomly distributed on the inner tube. The latter curve was calculated after averaging over ten different random distributions of the Stone-Wales pairs. It can be seen that three of the two major features of the bamboo junction are still visible even if traditional disorder is taken into account, suggesting that the effect should be visible in a transport measurement on a real sample that contains traditional lattice defects.

Figure 9

(Color online) Structure of the inner shell of a (6,4)@(10,10) DWCNT with eight bamboo junctions in the inner shell during a molecular-dynamics simulation at 3500 K using the Tersoff potential and the NVT Berendsen thermostat. Geometries taken at 0 ns, 0.01 ns, 0.2 ns, 0.5 ns, and 3 ns from top to bottom, respectively.