Cotunneling Drag Effect in Coulomb-Coupled Quantum Dots

In Coulomb drag, a current flowing in one conductor can induce a voltage across an adjacent conductor via the Coulomb interaction. The mechanisms yielding drag effects are not always understood, even though drag effects are sufficiently general to be seen in many low-dimensional systems. In this Letter, we observe Coulomb drag in a Coulomb-coupled double quantum dot and, through both experimental and theoretical arguments, identify cotunneling as essential to obtaining a correct qualitative understanding of the drag behavior.

Figure 1

Device and model. (a) Top-down SEM image of a device nominally identical to that measured. $\mathrm{Ti}/\mathrm{Au}$ gate electrodes (light gray) are patterned on the substrate surface (dark gray). Colored circles represent the QDs. Arrows indicate where electrons can tunnel. (b) Cartoon showing names of gates, reservoirs, and dots. ${\mathrm{\Gamma }}_{\mathit{S}\mathit{i}}$ is the tunnel rate between reservoir $\mathit{S}\mathit{i}$ and dot $i$. (c) Capacitor and tunnel junction network. Interdot tunneling is strongly suppressed and not included in the model. Direct capacitance between gate P1 (2) and dot 2 (1) is omitted from the diagram for clarity, along with some labels.

Figure 2

Coulomb drag. (a) Schematic charge stability diagram for a CC-DQD. Dots indicate triple points. Red (blue) solid lines are charge transitions for dot 1 (2). As $V_{S1}$ increases from zero, excited states appear in $G_1$ within shaded regions. Roman numerals are used later for reference. (b) Sum of measured conductances $G_i=d{I}_i/d{V}_{\mathit{S}\mathit{i}}$ for $V_{S1}=V_{S2}=0$, as a function of dot levels ${ϵ}_1$, ${ϵ}_2$. (c),(d) Measured $G_1$ (c) and $G_2$ (d) for $V_{S1}=0.5\mathrm{mV}$. (e),(f) Measured $I_2$ for $V_{S1}=0.5\mathrm{mV}$ (e) and $V_{S1}=-0.5\mathrm{mV}$ (f). In both cases the current $I_2$ flows in the same direction, is strongest in region (ii), and persists in regions (i) and (iii). Dashed white lines in (c) and (e) are discussed in the text.

Figure 3

For small drive bias $V_{S1}$, the drag current $I_2$ appears to vanish depending on the drag dot’s level. (a) $G_1$ for $-{ϵ}_2=0.12\mathrm{meV}$, on the border of region (ii) and (iii) in Fig. 2. The color scale is saturated to emphasize fine features. (b) Drag current $I_2$ for $-{ϵ}_2=0.12\mathrm{meV}$ persists even in the limit that drive bias $V_{S1}\to 0$. (c) $G_1$ for $-{ϵ}_2=0.47\mathrm{meV}$, well within region (iii) in Fig. 2. The color scale is saturated, and appears similar to (a). (d) $I_2$ for $-{ϵ}_2=0.47\mathrm{meV}$ (region (iii)]. Below $V_{S1}\sim 0.12\mathrm{meV}$, the drag current is unmeasurable. This gap also appears for $-{ϵ}_1<0.1$ and negative $V_{S1}$ (not shown), and is the same value within measurement accuracy.

Figure 4

Temperature and dot level dependence of drag current. Icons indicate where cuts are taken (as in Fig. 2). $V_{S1}=500\mu \mathrm{V}$, $V_{S2}=0$. (a) Temperature-dependent $I_2$. Top: $-{ϵ}_2=0.05\mathrm{meV}=\mathit{U}/2$ (middle of region (ii) in Fig. 2); bottom: $-{ϵ}_2=0.5\mathrm{meV}=5\mathit{U}$ (deep in region (iii)]. The drag current does not change appreciably from 20 to 155 mK in either case. (b) Drag current $I_2$ can be measured even when dot 2’s levels are far off resonance, provided a current is flowing in dot 1.

Figure 5

Theory of cotunneling leading to drag. (a) Illustration of a sequential process leading to drag current. An electron hops into the drag dot (dot 2). Next, an electron hops into the drive dot (dot 1), causing the dot 2 level to rise due to interdot Coulomb repulsion in turn allowing the electron in 2 to be transferred to the right. The whole process involves four tunneling rates and is hence of order ${\mathrm{\Gamma }}^4$. (b) Pure cotunneling process leading to drag current. Two electrons tunnel simultaneously onto the dots. They then tunnel off simultaneously. Since each cotunneling process has a probability ${\mathrm{\Gamma }}^2$, the cotunneling drag process is of order ${\mathrm{\Gamma }}^4$ as in (a). (c) Calculated drag current in units of $e\mathrm{\Gamma }/\hslash$ for drive voltage $V=0.5\mathrm{mV}$ and system parameters extracted from the experiment. (d),(e) Drag current as a function of drive dot level and $V$ for (d) ${ϵ}_2=-0.2\mathrm{meV}$ and (e) ${ϵ}_2=-0.4\mathrm{meV}$.