Correlated Coulomb Drag in Capacitively Coupled Quantum-Dot Structures

We study theoretically Coulomb drag in capacitively coupled quantum dots (CQDs)—a bias-driven dot coupled to an unbiased dot where transport is due to Coulomb mediated energy transfer drag. To this end, we introduce a master-equation approach that accounts for higher-order tunneling (cotunneling) processes as well as energy-dependent lead couplings, and identify a mesoscopic Coulomb drag mechanism driven by nonlocal multielectron cotunneling processes. Our theory establishes the conditions for a nonzero drag as well as the direction of the drag current in terms of microscopic system parameters. Interestingly, the direction of the drag current is not determined by the drive current, but by an interplay between the energy-dependent lead couplings. Studying the drag mechanism in a graphene-based CQD heterostructure, we show that the predictions of our theory are consistent with recent experiments on Coulomb drag in CQD systems.

Figure 1

(a) Charge stability diagram of two capacitively coupled QDs as a function of their gate detuning $\delta =V_2-V_1$ and common gate $\epsilon =V_1+V_2$. (b) Sequence of sequential and cotunneling processes underlying the drag mechanism in the vicinity of the triple points [closed circle in (a)]. Away from the triple points, the drag is driven by cotunneling only [arrow in (a)]. Energy-dependent lead couplings are essential for the mechanism to induce a directional current in the drag system. (c) Illustration of a graphene-based CQD heterostructure with two QDs defined in stacked graphene layers separated by a thin isolating dielectric [33]. A series of top and bottom gates control the potentials on the quantum dots ($V_{1/2}$) and their adjacent graphene leads.

Figure 2

Drive (top) and drag current (bottom) for the graphene-based CQD in Fig. 1, with the voltage configuration in Fig. 3. (a),(c) Current vs common gate and gate detuning with a bias voltage $e{V}_{\mathrm{sd}}=0.2$ applied to the drive QD. (b),(d) Bias dependence of the drive and drag currents at the gate voltages $(V_2-V_1,V_1+V_2)$ marked by dots in the left plots [red: (0.0,1.0), yellow: (0.0,0.8), green: (0.0,0.0), blue: $(-0.2,0.0)$]. Parameters (in units of $U_{12}$): $U_{12}=1$, ${\mathrm{\Gamma }}_{L_{1}0/R_{1}0}={\mathrm{\Gamma }}_{L_{2}0/R_{2}0}=0.01\equiv \mathrm{\Gamma }$, $\partial {\mathrm{\Gamma }}_{L_2}=-\partial {\mathrm{\Gamma }}_{R_2}$, $t_{L_2}=t_{R_2}$, $k_{B}T=0.01$.

Figure 3

(a) Energy level diagram of the graphene-based CQD in Fig. 1. The QD levels, ${ϵ}_i=-e{V}_i$, and the positions of the Dirac points in the leads, $E_{\alpha 0}=-e{V}_{\alpha }$, are controlled by local gates. (b) Drag current as a function of gate voltage on the leads of the drag system (see Dirac cone insets) at the upper triple point in the stability diagram. Parameters (in units of $U_{12}$): $U_{12}=1$, ${\mathrm{\Gamma }}_{L_{1}0/R_{1}0}=0.01\equiv \mathrm{\Gamma }$, ${\mathrm{\Gamma }}_{L_{2}0/R_{2}0}\propto |{\mu }_0-E_{L_{2}0/R_{2}0}|$, $\partial {\mathrm{\Gamma }}_{L_2/R_2}=\mathrm{sgn}({\mu }_0-E_{L_{2}0/R_{2}0})$, $t_{L_2}=t_{R_2}$, $e{V}_{\mathrm{sd}}=0.1$, $k_{B}T=0.01$.

Figure 4

Bias spectroscopy. The plots show the current through drive (top) and drag (bottom) QDs at the center of the 10,01 degeneracy line with the bias applied to the drive QD. (a),(c) Linear scale. (b),(d) Log scale. The dashed (dotted) lines mark the boundaries to the regions where the currents are dominated by nonlocal cotunneling (sequential tunneling). See Fig. 2 for parameters.