Energy dispersion in graphene and carbon nanotubes and molecular encapsulation in nanotubes

Density-functional calculations of electronic and vibrational dispersion energies for pristine graphite monolayer (graphene) and single-walled carbon nanotubes (SWCNTs) are presented. Optimized parameters for nonlocal norm-preserving pseudopotentials which replace the potential field due to core electrons are given. Comparison with observations, where available, is made. The effect of encapsulation of carbon nanotubes with an alkali-halide matrix is numerically investigated. The electronic band structure of encapsulated SWCNT is noticeably modified, as is its charge density, and hence its optical properties.

Figure 1

LCAO basis set convergence test performed for a (4,4) SWCNT. From a cost-benefit perspective, DZP produces the most accurate results. The labels stand for single (s), double (d), triple (t), $\zeta$ (z), and polarized (p) orbitals.

Figure 2

Electronic energy dispersion relations for a graphene sheet using (a) the SIESTA developers’ PP parameters and (b) this work. Troullier-Martin PP and GGA DFT flavor are employed in both cases. The mesh cutoff was $200\mathrm{Ry}$, with 243 $k$ points and a $50\mathrm{meV}$ PAO energy shift.

Figure 3

Electronic density of states for a graphene sheet using the (a) SIESTA developers’ PP parameters and (b) this work. A total of 3465 $k$ points were employed.

Figure 4

Electronic energy dispersion relations for an armchair (6,6) SWCNT using the (a) SIESTA developers’ PP parameters and (b) this work. Five $k$ points were employed and the mesh cutoff was at $200\mathrm{Ry}$.

Figure 5

Electronic energy dispersion curves for the zigzag (10,0) SWCNT using the optimized PP parameters.

Figure 6

Electronic density of states for the (10,0) zigzag SWCNT using the optimized PP parameters in (b) with 900 $k$ points. The mesh cutoff was kept at $400\mathrm{Ry}$ and the PAO energy shift was $50\mathrm{meV}$. The experimental observation of DOS, from Ref. 30, is also shown in (a).

Figure 7

Phonon dispersion modes in graphene. A total of 98 atoms, arranged in $3\times 3\times 0$ $(N_x,N_y,N_z)$, were used, where $N_x,N_y,N_z$ refer to the number of atoms on each side of the central carbon atom in the unit cell.

Figure 8

Naked (10,10) and filled KI@(10,10) nanotubes. The results are shown for a three-ring supercell containing 120 C atoms in (a) and an additional 4 I and 4 K atoms in (b). The Fermi crossing is now at $1∕3$ the distance from the $\Gamma$ point, at $k=\frac{2\pi }{9a}$.

Figure 9

(Color online) (10,10) SWCNT charge density in the absence of (top) and due to the presence of K and I ions. The I atom is in the plane and the K atom is above the plane. The charge density is modified.