Density functional calculations for ${\mathrm{C}}_{60}$ peapods

Nanotubes filled with ${\mathrm{C}}_{60}$ molecules, commonly referred to as peapods, are studied using the local density approximation of the density functional theory. The band structure of the peapods are investigated in detail, concentrating in particular on the ${\mathrm{C}}_{60}$ derived bands and their hybridization with the nanotube bands. It is found that the $t_{1u}$ states of fullerene are always located close to the Fermi level or midgap for metallic and insulating nanotubes, respectively. This is a consequence of the strong electron affinity of the ${\mathrm{C}}_{60}$ molecule. For insulating nanotubes, the $t_{1g}$ states, on the other hand, are located roughly in the energy region of the third van Hoove singularity, resulting in some degree of hybridization, if the corresponding nanotube states possess the proper symmetry. Except for narrow nanotubes, simulated scanning tunneling spectroscopy images of the peapods show a very low contrast for ${\mathrm{C}}_{60}$ derived states.

Figure 1

Binding energy of ${\mathrm{C}}_{60}$@(11,11) vs the position of the ${\mathrm{C}}_{60}$ molecule. The ${\mathrm{C}}_{60}$ molecule is moved from the center of the tube (initial position) towards the tube wall by an offset $x$.

Figure 2

Binding energy of ${\mathrm{C}}_{60}$ in various peapods. The energy difference $\Delta E$ between the ${\mathrm{C}}_{60}@(n,m)$ peapod and the free ${\mathrm{C}}_{60}$ and free $(n,m)$ nanotube are shown vs the nanotube diameter $d_t$. The circles show the results for the ${\mathrm{C}}_{60}$ placed in the center of the tube, while the diamonds are results for the eccentric placement of the ${\mathrm{C}}_{60}$. Circles correspond to the following tubes in the specified order: (15,0), (9,9), (12,6), (16,0), (17,0), (10,10), (18,0), (16,4), (14,7), (19,0), and (11,11).

Figure 3

Band structure of metallic (10,10), (11,11), (16,4), and (18,0) peapods. The Fermi level is marked by a dashed line. The $t_{1u}$ orbitals are indicated by full drawn lines, The filled circles mark the states localized on ${\mathrm{C}}_{60}$, with the size of the circles being a measure of the degree of localization. The other states are shown by small plus symbols connected by dotted lines, to guide the eye (the dotted lines are not necessarily linking the bands properly).

Figure 4

Zone-folding in zigzag [(5,0), (6,0)] and armchair [(5,5), (6,6)] tubes. The hexagons indicate the Brillouin zone of a graphite layer. Solid horizontal lines map to the line $\Gamma ,Z$ in the nanotube. Points that map to $\Gamma$ in the nanotube are indicated by open circles and labeled according to the symmetry the corresponding electronic states will have in the nanotube. Gray dashed lines are guidelines to the eye.

Figure 5

Schematic band structure for the metallic (10,10) peapod. Dashed lines represent states before hybridization, whereas hybridized bands are indicated by full lines. States marked with $M$ are nanotube derived states with an azimuthal quantum number of $M=0,\pm 1$.

Figure 6

Band structure for semiconducting zigzag (16,0) and (17,0) peapods and for the chiral semiconducting tube (14,7). The filled circles mark the states localized on ${\mathrm{C}}_{60}$.

Figure 7

Electronic DOS for the (13,6) peapod filled with (a) six and (b) one ${\mathrm{C}}_{60}$ molecules. DOS for empty tube is shown as reference (dashed line). The DOS has been broadened by a Gaussian with a width of $0.1\mathrm{eV}$. van Hoove singularities are marked by arrows.

Figure 8

Simulated scanning tunneling spectra $\left(dI∕dV\right)$ for (a) (16,0), (b) (17,0), and (c) (10,10) peapods as the tip is moved along the tube axis $\left(z\right)$. Bright areas correspond to stronger current increase $\left(dI∕dV\right)$ whereas dark areas correspond to no current increase. The images are four (five) times repeated along the peapod translational period $\left(z\right)$ for the zigzag (armchair) tubes. Positions of the fullerenes are indicated by white lines.

Figure 9

Simulated scanning tunneling images for (13,6) peapod, filled with (a) six and (b) one ${\mathrm{C}}_{60}$ molecules, respectively.