Transport properties of double-walled carbon nanotube quantum dots

The transport properties of quantum dot (QD) systems based on double-walled carbon nanotubes (DWCNTs) are investigated. The interplay between microscopic structure and strong Coulomb interaction is treated within a bosonization framework. The linear and nonlinear current-voltage characteristics of the QD system are calculated by starting from the Liouville equation for the reduced density matrix. Depending on the intershell couplings, an eight-electron periodicity of the Coulomb blockade peak spacing in the case of commensurate DWCNT QDs and a four-electron periodicity in the incommensurate case are predicted. The contribution of excited states of DWCNTs to the nonlinear transport is investigated as well.

Figure 1

(Color online) Schematic experimental setup of a DWCNT QD system. A finite length DWCNT is deposited on a substrate and weakly connected to the source and drain leads through its outer shell. A gate electrode is capacitively coupled to the DWCNT QD and controls the electrochemical potential in the QD. The dashed lines denote the inner shell in the DWCNT.

Figure 2

(Color online) Energy spectra of metallic SWCNT and DWCNT with PBCs. (a) Energy spectrum of a metallic SWCNT. There are two Fermi points and two branches $L∕R$ with left-moving (right-moving) electrons at each Fermi point. (b) Energy spectrum of an i-DWCNT. It consists of the energy spectra of the outer and inner graphene shells (±), which are not coupled to each other because of the vanishing intershell coupling, cf. Eq. (12). (c) Energy spectrum of a c-DWCNT. Because of the finite intershell coupling, it is composed of the bonding and antibonding bands (±), which are shifted vertically along the $\epsilon$ axis, cf. Eq. (15).

Figure 3

(Color online) Cross section of a DWCNT. Atoms $A$ and $B$ in two shells of radii $R_{+}$ and $R_{-}$, respectively, are projected onto this cross section. Such atoms are described by the coordinates $(u_{+},v_{+})$ and $(u_{-},v_{-})$, where $u_{\pm }$ are along the tube axis and $v_{\pm }$ measure the atom positions on the outer (inner) circumference.

Figure 4

(Color online) Energy spectra of metallic SWCNT and DWCNT with OBCs. (a) Energy spectrum of a metallic SWCNT. There are two branches $\tilde{L}∕\tilde{R}$ with left (right) moving electrons. The parameter ${\epsilon }_0$ is the level spacing and $\Delta$ describes the mismatch of the Fermi point, cf. Eq. (10). (b) Energy spectrum of an i-DWCNT and (c) of a c-DWCNT. The parameter ${\Delta }_{\pm }$ describes the mismatch of the Fermi point in the shell ± and for a c-DWCNT is ${\Delta }_{+}={\Delta }_{-}=\Delta$. The parameter $\zeta$ describes the mismatch of the states in two bands, cf. Eq. (16).

Figure 5

(Color online) Calculated linear conductances as a function of the gate electrochemical potential in c-DWCNT QD systems with different parameters. (a) $\Delta =\zeta =0.0$, $W_{00}^{++}=W_{00}^{--}=W_{00}^{+-}∕2=5.0{\epsilon }_0$, and $k_{B}T=0.025{\epsilon }_0$, where the level spacing ${\epsilon }_0$ is used as the unit of energy. The coupling strengths are ${\gamma }_{s\pm }={\gamma }_{d\pm }=0.02{\epsilon }_0$. (b) $\Delta =0.2$ and $\zeta =0.3$. The remaining parameters are the same as those in (a). In both cases, the linear conductances in c-DWCNTs show an eight-electron periodicity. In the case of zero mismatch parameters shown in (a), an eight-electron periodicity of the heights of the conductance peaks also occurs. For finite mismatch shown in (b), the peak heights are equal but the eight-electron periodicity of the addition energies, i.e., the peak distances, remains (to emphasize this, we assign to each group of eight peaks different colors).

Figure 6

(Color online) Calculated linear conductances as a function of the gate electrochemical potential in i-DWCNT QD systems with different parameters. (a) ${\Delta }_{+}={\Delta }_{-}=0.0$, $W_{00}^{++}=5{\epsilon }_0$, $W_{00}^{--}=6.0{\epsilon }_0$, $W_{00}^{+-}=5.5{\epsilon }_0$, and $k_{B}T=0.025{\epsilon }_0$, where the level spacing ${\epsilon }_0$ is used as the unit of energy. The coupling strengths are ${\gamma }_{s+}={\gamma }_{d+}=0.02{\epsilon }_0$ and ${\gamma }_{s-}={\gamma }_{d-}=0$. (b) ${\Delta }_{+}=0.2$ and ${\Delta }_{-}=0.3$. The remaining parameters are the same as those in (a). The linear conductances in i-DWCNTs show a four-electron periodicity. In the absence of mismatch shown in (a), the four-electron periodicity is observed also in the peak heights, while it is no longer observed at finite mismatch shown in (b). However, the addition energy, i.e., the peak distance, shows a four-electron periodicity in both cases (to emphasize this, we assign to each group of four peaks different colors).

Figure 7

(Color online) Calculated stability diagrams of DWCNT QD systems. (a) Stability diagram of a c-DWCNT QD. The parameters are $\Delta =0.2$, $\zeta =0.3$, $W_{00}^{+}=W_{00}^{-}=W_{00}^{+-}∕2=5.0{\epsilon }_0$, and $k_{B}T=0.05{\epsilon }_0$, where the level spacing ${\epsilon }_0$ is used as the unit of energy. We use ${\gamma }_{s\pm }={\gamma }_{d\pm }=0.02{\epsilon }_0$ for the coupling strengths between the leads and the DWCNT QD. (b) Stability diagram of an i-DWCNT QD. The parameters are ${\Delta }_{+}=0.2$, ${\Delta }_{-}=0.3$, $W_{00}^{+}=5{\epsilon }_0$, $W_{00}^{-}=6.0{\epsilon }_0$, $W_{00}^{+-}=5.5{\epsilon }_0$, and $k_{B}T=0.05{\epsilon }_0$. The coupling strengths are ${\gamma }_{s+}={\gamma }_{d+}=0.02{\epsilon }_0$ and ${\gamma }_{s-}={\gamma }_{d-}=0$. The size of the Coulomb diamonds shows an eight-electron periodicity in c-DWCNT QDs, as shown in (a), while it shows a four-electron periodicity in i-DWCNT QDs, as shown in (b).