Transport properties of double quantum dots with electron-phonon coupling

We study transport through a double quantum dot system in which each quantum dot is coupled to a phonon mode. Such a system can be realized, e.g., using a suspended carbon nanotube. We find that the interplay between strong electron-phonon coupling and interdot tunneling can lead to a negative differential conductance at bias voltages exceeding the phonon frequency. Various transport properties are discussed, and we explain the physics of the occurrence of negative differential conductance in this system.

Figure 1

A carbon nanotube is suspended between two metallic leads (blue) held at chemical potentials ${\mu }_L$ and ${\mu }_R$. The additional support in the center separates the nanotube into two regions; each free to oscillate. Tunnel barriers are depicted in gray. The orange spheres denote quantum dots that form on each suspended part of the nanotube.

Figure 2

Schematic picture of the setup described by the Hamiltonian (1). Two quantum dots with onsite energies ${\xi }_L$ and ${\xi }_R$ are coupled to each other with a tunneling amplitude $t_D$, and to electron reservoirs on the left and right with tunneling amplitudes $t_L$ and $t_R$, respectively. The electron reservoirs are normal-metal leads with chemical potentials ${\mu }_L$ and ${\mu }_R$, respectively. A phonon mode with frequency ${\Omega }_L$ (${\Omega }_R$) is coupled to the left (right) dot.

Figure 3

Current $I$ through the double dot system as a function of bias voltage $V$ for different values of electron-phonon coupling $g$ and symmetric level energies ${\tilde{\xi }}_L={\tilde{\xi }}_R$. The current decreases with increasing electron-phonon coupling. For a fixed (nonzero) value of the electron-phonon coupling, the current also decreases (in some regions) with increasing bias voltage. Parameters are given in the legend. Here and in the following figures, energies are in units of $\Omega$. Hence, the bias voltage $V$ is shown in units of $\Omega /e$ and the current $I$ in units of $(e/h)\Omega$.

Figure 4

Differential conductance $dI/dV$ for the same parameters as in Fig. 3. We clearly see the negative differential conductance.

Figure 5

Current $I$ through the double dot system as a function of bias voltage $V$ for different electron-phonon couplings $g$ and an asymmetric choice of level energies ${\tilde{\xi }}_L\ne {\tilde{\xi }}_R$. In comparison to Fig. 3, the current for a fixed electron-phonon coupling is now increasing with the bias. All parameters are given in the legend.

Figure 6

Here, we show the differential conductance for the same parameters as in Fig. 5. We clearly see the absence of a negative differential conductance.

Figure 7

The function $\mathcal{V}$ as a function of $V$ for ${\tilde{\xi }}_L={\tilde{\xi }}_R=1$ (black) and ${\tilde{\xi }}_L=1$, ${\tilde{\xi }}_R=2$ (blue). For ${\tilde{\xi }}_L={\tilde{\xi }}_R=1$, $\mathcal{V}$ decreases at every phonon sideband in contrast to the case ${\tilde{\xi }}_L=1$, ${\tilde{\xi }}_R=2$. In this case at every phonon sideband (the first one being at $V\approx 4$) $\mathcal{V}$ increases. (Note the blue curve is shifted to the left due to the asymmetry in the setup.)

Figure 8

Diagonal elements of the density matrix ${\rho }_{00}$ (black), ${\rho }_{LL}$ (blue), and ${\rho }_{RR}$ (red). In b) $g=0$ and level energies are chosen symmetrically. In (a) and (c), $g=0.8$ and level energies are chosen symmetric and asymmetric, respectively. Here, $t_D=0.4$ and ${\tilde{\xi }}_L=1$.