Andreev-Coulomb Drag in Coupled Quantum Dots

The Coulomb drag effect has been observed as a tiny current induced by both electron-hole asymmetry and interactions in normal coupled quantum dot devices. In the present work we show that the effect can be boosted by replacing one of the normal electrodes by a superconducting one. Moreover, we show that at low temperatures and for sufficiently strong coupling to the superconducting lead, the Coulomb drag is dominated by Andreev processes, is robust against details of the system parameters, and can be controlled with a single gate voltage. This mechanism can be distinguished from single-particle contributions by a sign inversion of the drag current.

Figure 1

Superconductivity-induced Coulomb drag in hybrid dot structures. (a) Scheme of the capacitively coupled double quantum dot device analyzed in this work. A drag current is generated in the passive dot ($p$) connected to unbiased normal ($N$) and superconducting ($S$) electrodes due to the Coulomb interaction with electrons tunneling through the active dot $a$ attached to voltage biased terminals ($L$ and $R$). In this diagram, an electron in $N$ is transformed into a Cooper pair in $S$ with a retroreflected hole. This Andreev process is correlated through the interaction (red line) with a charge fluctuation in the active subsystem (blue arrow). (b) Drag current $I_{\mathrm{drag}}$ as a function of the passive dot level ${ϵ}_p$ and temperature $T$ (normalized with the critical temperature $T_c$). Cooper pairs and quasiparticles contribute with opposite signs. At low temperature, the current is given by Andreev processes (lower insets). By increasing temperature, a crossover occurs where they coexist with quasiparticle tunneling (upper insets). Parameters (in meV): $\mathrm{\Delta }(T=0)=0.2$, ${\mathrm{\Gamma }}_S={\mathrm{\Gamma }}_N={\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=0.05$, $V_{\mathrm{bias}}=5$, ${ϵ}_a=0$, and $U_p=5U_{ap}=0.5$.

Figure 2

Andreev-Coulomb drag mechanism. (a) Pairing induces avoided crossings in the even states $|\pm ,n⟩$. Two cyclic processes are highlighted which correspond to Eq. (2), with $n=1$, $n^{\prime }=0$ (orange) and $n=0$, $n^{\prime }=1$ (green). These cycles are assisted by charge fluctuations in the active dot and lead to finite drag currents. (b) The imbalance of tunneled electrons and holes dictates the sign of the current by means of the asymmetry of the transition rates, parametrized here by $\mathcal{R}=(r_{+0}{r}_{-1}-1)/(r_{+0}{r}_{-1}+1)$, depending on the level position ${ϵ}_p$. The gray line in (a) and (b) marks the electron-hole symmetry point ${ϵ}_p^{\mathrm{eh}}$. Parameters are as in Fig. 1.

Figure 3

Unraveling the Andreev and quasiparticle contributions to the drag current. (a) Drag current as a function of ${\mathrm{\Gamma }}_S$ and ${ϵ}_p$. (b) Different contributions from the single quasiparticle ($I_{\mathrm{qp}}$) and pair tunneling ($I_A$) currents to the total drag current ($I_t$). Dashed lines in (b) are fitting curves to the ${\mathrm{\Gamma }}_S$ and ${\mathrm{\Gamma }}_S^2$ lines. Parameters are as in Fig. 1, except for $T=2T_c/3$.

Figure 4

Drag and drive currents as functions of experimentally tunable gate and bias voltages. (a),(b) Mutual backaction of the drag and drive systems as ${ϵ}_a$ and ${ϵ}_p$ scan the degeneracy points of the stability diagram for $k_{B}T=0.01\mathrm{meV}$ and $V_{\mathrm{bias}}=0.6\mathrm{meV}$. (c),(d) The drag current does not change sign at zero bias voltage, proving its nonlinearity, here for ${ϵ}_p=0$. (c) At low temperatures, the Andreev-Coulomb drag current is blocked at low voltages. Lifting of the blockade affects the drive response. (d) At higher temperatures, these features are washed out. Quasiparticle contributions change the sign of the current. Other parameters are as in Fig. 1 and $k_B{T}_c=0.15$ meV.