Selective focusing of electrons and holes in a graphene-based superconducting lens

We show that a graphene pnp junction with a central superconducting electrode acts as a Veselago lens for incoming electrons by focusing them and their phase-conjugated counterpart (holes) into different points of the optical axis. This selective focusing suggested by a simple trajectory analysis is confirmed by fully microscopic calculations. Although the focusing pattern is degraded by deviations from the ideal conditions, we show that it remains visible for a wide range of parameters. We discuss how this property can be useful for the detection of entangled electron pairs.

Figure 1

Graphene superconducting pnp junction. The band structure of each region is shown on top for the case when the normal electrodes $L$ and $R$ are adjusted to have the opposite doping level than the superconducting central region $S$. Incoming electrons from the left electrode (solid red lines) transform into evanescent electron and hole-like excitations inside the superconductor (dashed red and blue lines, respectively). In the right region, electrons and holes (solid blue lines) focus into distinct regions.

Figure 2

(a) Map of the electron transmission $T_{ee}(x,y)$ near the focusing point $x=3W_S/2$. The width of the superconducting region is $W_S=\xi \sim 1\mu$m and the doping levels are $|E_F^{L,S,R}|=0.5$ eV. (b) For the same parameters, map of the hole transmission $T_{eh}(x,y)$ near the focusing point $x=5W_S/2$. (c) Sketch of the trajectories of electrons and holes through the superconductor into the region $R$. See the text for details. (d) Logarithmic scale plot of the transmission of electrons (red) and holes (blue) into region $R$ for $y=0$, showing the profiles of (a) and (b) where the peaks due to internal reflections can be distinguished.

Figure 3

Microscopic calculation for $T_{ee}(x,0)$ and $T_{eh}(x,0)$ when the injection of electrons is done at $d=-3W_S/2$. The inset shows a sketch of the classical trajectories. Notice that the peak of $T_{eh}$ at $x=3.5W_S$ is $\sim 5$ times larger than the EC background.

Figure 4

(a): Map of $T_{eh}(x,y)$ for an asymmetric pnp junction. We set $E_F^R=0.4$ eV and keep the rest of the parameters the same as in Fig. 3, where $d>W_S$. The refracted waves in the right region form a characteristic interference pattern showing two caustic curves near the cusp for each of the transmitted holes. (b): Map of $T_{eh}(x,y)$ for a pnp junction calculated using the TB model. The length of the central superconducting region is $W_S=85.2$ nm. The gate potentials used are $E_F^{L,R}=-E_F^S=0.6$ eV, with $\Delta =2.7$ meV. The left panel includes no disorder. The central panel has $V_0=54$ meV. The right panel has $V_0=81$ meV. The smearing of the potential for the three panels has a range of $\sim 35$ nm.