Carbon Nanotubes as Cooper-Pair Beam Splitters

We report on conductance measurements in carbon nanotube based double quantum dots connected to two normal electrodes and a central superconducting finger. By operating our devices as beam splitters, we provide evidence for crossed Andreev reflections tunable in situ. This opens an avenue to more sophisticated quantum opticslike experiments with spin entangled electrons.

Figure 1

(a) SEM image of a typical Cooper-pair splitter device in false colors with the two biasing schemes sketched. The bar is $1\mu \mathrm{m}$. A central superconducting electrode is connected to two quantum dots engineered in the same single wall carbon nanotube (in purple [lightest gray]) which bridges between electrodes $L$ and $R$. (b) The elementary processes which carry current in the superconducting ($S$) state. In addition to the conventional local Andreev reflection process, the crossed Andreev reflection can occur in which a Cooper pair is split in the two quantum dots. The relative probability of each of these processes can be inferred from the topology of the beam splitter.

Figure 2

(a) Color scale plot of the differential conductance $G_L$ as a function of side gate voltage 1 $V_{\mathrm{SG}1}$ and side gate voltage 2 $V_{\mathrm{SG}2}$. The dc bias voltage is zero here and the ac modulation $1.8\mu \mathrm{V}$. (b) Differential conductance (black circles with lines) measured out of resonance in the middle injection scheme at 100 mK and 0 mT. In green (light gray) squares, the differential conductance at 44.5 mT. In red (medium gray) solid lines, the BCS fit which yields an energy gap $\Delta$ of $85\mu \mathrm{eV}$ and an electronic temperature of 100 mK.

Figure 3

(a) Focus on a specific region for the differential conductance $G_L$ as a function of $V_{\mathrm{SG}1}$ and $V_{\mathrm{SG}2}$ (AC1). The yellow arrow indicates the direction in which the linear scan of panel (b) has been taken. (b) Measurements of the conductance in the normal state (black solid lines) and in the superconducting state (red [medium gray] solid lines) for $G_R$ and $G_L$ along the direction of the yellow arrow. For the sake of clarity, the $N$ state conductances have been multiplied by $1/3$. The model calculations are in green [dark gray] solid lines for the $S$ state and in orange [light gray] solid lines for the $N$ state. In blue dashed lines, the CAR probability. (c) Similar graph as in (b) for anticrossing 2 (AC2).

Figure 4

Bottom panel: Asymmetry in the superconducting state versus asymmetry in the normal state for 31 out of the 35 anticrossings studied (red filled squares) and 11 resonance peaks away from the anticrossings (black stars). The dashed curve corresponds to ${\alpha }_S={\alpha }_N^2$. The dashed line corresponds to ${\alpha }_S={\alpha }_N$. The green [medium gray] solid line is the full theory (see20) and the pink [light gray] solid line is the full theory without the CAR process. Top panel: Efficiency of the Cooper-pair splitting for the device as deduced from the values of ${\alpha }_S$ (see text). In dashed lines, the theoretical curve.