Transport through multiply connected quantum wires

We study transport through multiply coupled carbon nanotubes (quantum wires) and compute the conductances through the two wires as a function of the two gate voltages $g_1$ and $g_2$ controlling the chemical potential of the electrons in the two wires. We find that there is an equilibrium cross-conductance, and we obtain its dependence on the temperature and length of the wires. The effective action of the model for the wires in the strong coupling (equivalently Coulomb interaction) limit can also be mapped to a system of capacitively coupled quantum dots. We thus also obtain the conductances for identical and nonidentical dots. These results can be experimentally tested.

Figure 1

Carbon nanotube wires 1 and 2 are stretched between the source and the drain. $G_{\mathrm{A}}$ and $G_{\mathrm{B}}$ are the two floating gates which generate a strong capacitive coupling between the two wires at the two ends.

Figure 2

Schematic diagram of the experimental setup. The wires are modeled as LLs with $K_{i=1,2}$, with the leads having $K_L=1$. The density-density coupling between the wires is denoted by its strength $\lambda$ and the two gate voltages controlling the densities of electrons in the wire are denoted by $g_1$ and $g_2$.

Figure 3

Schematic diagram of the wires.

Figure 4

Conductances in the plane of the two gate voltages $g_i$. For ${\mathcal{U}}_1={\mathcal{U}}_2$, the solid lines represent semi-maxima (“$+$” or “$-$” wire at resonance) and the crossings represent the maxima (both wires at resonance). The dotted line represents semimaxima (“$+$” or “$-$” wire at resonance) for ${\mathcal{U}}_1\ne {\mathcal{U}}_2$. There is no resonance at the crossings of the dotted lines and that region has been left blank.

Figure 5

The dependence of the low temperature $\left(T⪡T_d\right)$ cross-conductance (in units of $2e^2∕h$) on $T$ and $T_d$ for the case of $K=0.5$. Here $T_d=\hslash v∕k_{B}d$ is the temperature equivalent of the length $d$ of the wire. $\Lambda$ is the high energy cutoff scale. The overall scale of the conductance has been adjusted by adjusting $\alpha$, to be within the perturbative regime. Hence, it is only the power law which is significant.

Figure 6

The dependencies of the low temperature $\left(T⪡T_d\right)$ conductance (in units of $2e^2∕h$) on $T$ and $T_d$ for the case of $K=0.25$.

Figure 7

Schematic diagram of (a) tunnel coupled quantum dots in series and (b) capacitively coupled quantum dots in parallel.

Figure 8

Conductances in the plane of the gate voltages $g_1$ and $g_2$ for (a) tunnel coupled dots and (b) capacitively coupled dots. In the absence of coupling between the dots, the resonance maxima form a square grid in both cases as shown by the open circles. In (a), the open circles are the only points where there is a maxima, because both dots need to be at resonance. In (b), the solid lines indicate semimaxima (where one of the dots is on resonance) and the open circles denote maxima where both dots are at resonance. When interdot coupling is introduced, in (a), each point of resonance splits into two as shown by the solid circles. In (b), the lines of semimaxima shift as shown by the dotted lines, and there are no points where both dots are at resonance.