Band Structure and Quantum Conductance of Nanostructures from Maximally Localized Wannier Functions: The Case of Functionalized Carbon Nanotubes

We have combined large-scale, $\Gamma$-point electronic-structure calculations with the maximally localized Wannier functions approach to calculate efficiently the band structure and the quantum conductance of complex systems containing thousands of atoms while maintaining full first-principles accuracy. We have applied this approach to study covalent functionalizations in metallic single-walled carbon nanotubes. We find that the band structure around the Fermi energy is much less dependent on the chemical nature of the ligands than on the $s{p}^3$ functionalization pattern disrupting the conjugation network. Common aryl functionalizations are more stable when paired with saturating hydrogens; even when paired, they still act as strong scattering centers that degrade the ballistic conductance of the nanotubes already at low degrees of coverage.

Figure 1

MLWFs obtained from the disentanglement and localization procedure. We show in (a) the 3 inequivalent MLWFs obtained for a pristine metallic (5,5) CNT, clearly corresponding to a chemical picture of $p_z$ and $s{p}^2$ orbitals, and in (b) the same MLWFs for the case when an aminophenyl group (1% coverage) has been covalently attached.

Figure 2

Band structure, quantum conductance, and density of states of a (5,5) CNT as calculated in the present approach based on MLWFs (solid lines), and compared with the results of a full diagonalization (circles). The band structure has been unfolded over the BZ corresponding to the primitive 20-atom unit cell; the five vertical dashed lines indicate the 5 $\mathbf{k}$ points corresponding to $\Gamma$ in a 100-atom supercell.

Figure 3

$\Delta E_2$ in eV as a function of the position of the functional group ${\mathrm{FG}}_2$ when the group ${\mathrm{FG}}_1$ is attached in the position marked by × ($\Delta E_1$ is indicated in parenthesis) on a (5,5) CNT. ${\mathrm{FG}}_1$ and ${\mathrm{FG}}_2$ are, respectively, (a) H and H, (b) nitrophenyl and H, (c) aminophenyl and H, (d) phenyl and phenyl (in our calculation, the nitro and amino residues are in the para position). The arrow lies parallel to the nanotube axis, and we denoted with $\alpha$, $\beta$, $\gamma$, and $\delta$ the arrangements for four of the most stable pairings.

Figure 4

Band structure of a (5,5) CNT decorated with a periodic array of functionalization pairs in position $\alpha$ (1 pair every 100 carbons): (a) H pair (dots are from a Quantum-ESPRESSO calculation [12], for comparison), (b) nitrophenyl and H, (c) aminophenyl and H, (d) model calculation, where the 2 $p_z$ MLWFs on the functionalized sidewall carbons have been removed from the calculation, (e) pristine nanotube.

Figure 5

Top: infinite metallic (5,5) CNT functionalized in its central region by an array of phenyl pairs. Left panel: quantum conductance of an infinite (5,5) CNT with one isolated pair of ligands in positions $\alpha$ (a), $\beta$ (b), $\gamma$ (c), $\delta$ (d), or with a single ligand attached (e) (dashed line: ligands are hydrogen atoms; solid line: model calculation; dotted line: pristine CNT). Right panel: random distributions of pairs of ligands or of single ligands (arranged as in the left panel), for the case of 10 defects per 1000 carbons (solid line) or 30 per 3000 (dashed line), averaged over 5 random configurations.