Microscopic theory of Cooper pair beam splitters based on carbon nanotubes

We analyze microscopically a Cooper pair splitting device in which a central superconducting lead is connected to two weakly coupled normal leads through a carbon nanotube. We determine the splitting efficiency at resonance in terms of geometrical and material parameters, including the effect of spin-orbit scattering. While the efficiency in the linear regime is limited to $50\%$ and decays exponentially as a function of the width of the superconducting region, we show that it can rise to $\sim$$100\%$ in the nonlinear regime for certain regions of the stability diagram.

Figure 1

Schematic representation of local (a) and nonlocal (b) Andreev processes in a generic Cooper pair splitter. (c) Specific geometry considered in this work: a finite SWCNT coupled to normal leads at its ends and a central superconducting lead. (d) illustrates the potential profile along the tube.

Figure 2

Left panel: Conductance map for a metallic tube ($N=12$) in the normal state for the $p$-$n$-$p$ region with (lower) and without (upper) SO interactions. Right panel: Same for the semiconducting tube ($N=11$) but in a logarithmic scale. The geometrical parameters are $W=W_{L,R}=170$ nm.

Figure 3

Evolution of the CAR probability (blue or dark gray) and the splitting efficiency $\eta$ (green or light gray) in the linear regime as a function of the length $W$ of the central electrode for a metallic SWCNT with $N=12$ (upper panel) and a semiconducting one with $N=11$ (lower panel). The gate potentials $V_{gL,gR}$ are fixed at the points indicated by the circles in Fig. 2. The dashed lines represent the decay of the CAR probability for a 3D bulk superconductor multiplied by a factor ${10}^3$.

Figure 4

Color map of the splitting efficiency within the minimal model with parameters corresponding to a semiconducting tube with $W\sim 700$ nm (indicated by the arrow in Fig. 3) in the linear (a) and nonlinear (b) regimes. In the latter case the dot levels are varied along the line $E_L\sim -E_R$ indicated by the dashed red line of (a). The white dashed lines indicate the maxima in the spectral density, which is shown in (c) for the two dots along this line. Local and nonlocal Andreev processes at finite $V$ are indicated by the black arrows.

Figure 5

Evolution of the splitting efficiency $\eta$ as a function of the length $W$ of the central electrode for $N=11$. Different values of the parameter $\alpha$ are shown, representing different electrostatic potential profiles. The asymptotic behavior $\eta \sim 1/[1+\exp 2W/\xi \left(q\right)]$ is indicated by the black dashed line. The gate potentials $V_{gL,gR}$ are fixed at the values indicated by the centers of the circles of Fig. 2 in the main text.