Single-wall nanotubes: Atomiclike behavior and microscopic approach

Recent experiments about the low temperature behavior of a single-wall carbon nanotube (SWCNT) showed typical Coulomb blockade peaks in the zero bias conductance and allowed us to investigate the energy levels of interacting electrons. Other experiments confirmed the theoretical prediction about the crucial role which the long range nature of the Coulomb interaction plays in the correlated electronic transport through a SWCNT with two intramolecular tunneling barriers. In order to investigate the effects on low dimensional electron systems due to the range of electron electron repulsion, we introduce a model for the interaction which interpolates well between short and long range regimes. Our results could be compared with experimental data obtained in SWCNTs and with those obtained for an ideal vertical quantum dot. For a better understanding of some experimental results we also discuss how defects and doping can break some symmetries of the band structure of a SWCNT.

Figure 1

The dispersion relation and the quantized levels. The boxes in the figure represent energy levels and can be filled by a pair of electrons with opposite spins. (a) The general case without any degeneracy. (b) The fourfold degeneracy case with ${\delta }_{SM}=0$. (c) The eightfold degeneracy case. (d) Differences in the splitting due to the comparison between ${\delta }_{SM}$ and $\mathrm{\Delta }\epsilon$.

Figure 2

On the right we show analytical Aufbau results for the addition energy versus the number of electrons of a fourfold degeneracy model corresponding to different values of the range $r$ ($r=0$ dark gray dashed line, $r=0.4$ black line, and $r=1$ gray line). We show how the damping in the oscillations is due to a long range interaction while it does not appear for a $r=0$ model. Our predictions can be compared with the measured addition energy in the Cobden Nygard experiment displayed on the left.

Figure 3

(Color online) The asymmetric model calculation for the COs (conductance vs gate voltage) at different temperatures expressed in terms of $\mathrm{\Delta }$ calculated following the classical CB theory. Theoretical calculations show that the fine structure peaks are appreciable just for very low temperatures.

Figure 4

In (a), (b), and (c) results from three different models of interacting electrons in QD’s are displayed. The simple HF calculation in (a) has to be corrected because it does not take into account the classical capacitive effect. In order to determine it we recall that the electrons in the dot give a charge droplet of radius $R_D$ not fixed, so that we can approximate it with a disk of capacitance $C=C_0{R}_D$. The value of $R_D$ can be calculated from classical equations, $R_D\propto \sqrt{N+1}$. So we add the classical term to the damped oscillations due to long range affected Coulomb exchange and obtain the (b) and (c) plot in the figure.