Addition Energies of Fullerenes and Carbon Nanotubes as Quantum Dots: The Role of Symmetry

Using density-functional theory calculations, we investigate the addition energy (AE) of quantum dots formed of fullerenes or closed single-wall carbon nanotubes. We focus on the connection between symmetry and oscillations in the AE spectrum. In the highly symmetric fullerenes the oscillation period is large because of the large level degeneracy and Hund’s rule. For long nanotubes, the AE oscillation is fourfold. Adding defects destroys the spatial symmetry of the tubes, leaving only spin degeneracy; correspondingly, the fourfold behavior is destroyed, leaving an even/odd behavior which is quite robust. We use our symmetry results to explain recent experiments.

Figure 1

Addition energy (AE) spectra of the fullerenes ${\mathrm{C}}_{60}$, ${\mathrm{C}}_{76}$, ${\mathrm{C}}_{240}$, and ${\mathrm{C}}_{540}$. Electronic temperatures and basis sets adopted are indicated. Inset: the levels near the neutrality point, showing the large degeneracy factors. Note the lack of any even/odd oscillation in the AE; the large values occur upon completion of a shell.

Figure 2

AE spectra of the closed SWCNTs (a) ${\mathrm{C}}_{250}$ and ${\mathrm{C}}_{294}$, (b) ${\mathrm{C}}_{510}$ and ${\mathrm{C}}_{1010}$, and (c) ${\mathrm{C}}_{640}$. Note the large even/odd oscillation near the neutrality point in the small tubes and the fourfold oscillation for the longer tubes. Values of $J/\Delta$ (ratio of exchange shift to mean-level spacing) are calculated by averaging over the levels indicated by the arrows.

Figure 3

Density of states of an infinite $(5,5)$ nanotube. The low value at the neutrality point implies a large mean-level spacing in the closed tubes.

Figure 4

AE spectra of the ${\mathrm{C}}_{510}$ tube with three (5-7-7-5) defects separated widely and orientated randomly. 1, 2, 3, and 4 denote four different defect configurations. The presence of defects destroys the fourfold periodicity, but a strong even/odd effect remains.