Superconducting electron and hole lenses

We show how a superconducting region (S), sandwiched between two normal leads (N), in the presence of barriers, can act as a lens for propagating electron and hole waves by virtue of the so-called crossed Andreev reflection (CAR). The CAR process, which is equivalent to Cooper pair splitting into two N electrodes, provides a unique possibility of constructing entangled electrons in solid state systems. When electrons are locally injected from an N lead, due to the CAR and normal reflection of quasiparticles by the insulating barriers at the interfaces, sequences of electron and hole focuses are established inside another N electrode. This behavior originates from the change of momentum during electron-hole conversion beside the successive normal reflections of electrons and holes due to the barriers. The focusing phenomena studied here are fundamentally different from the electron focusing in other systems, such as graphene p-n junctions. In particular, due to the electron-hole symmetry of the superconducting state, the focusing of electrons and holes is robust against thermal excitations. Furthermore, the effects of the superconducting layer width, the injection point position, and barrier strength are investigated on the focusing behavior of the junction. Very intriguingly, it is shown that by varying the barrier strength, one can separately control the density of electrons or holes at the focuses.

Figure 1

A schematic view of the superconducting lens is shown beside the classical trajectories of electrons (solid blue lines) and holes (dashed red lines) originating from a point source inside ${\mathrm{N}}_1$. The focusing effect is caused by the crossed Andreev reflection (nonlocal electron-hole conversion) in combination with normal reflections which take place at the interfaces. It must be noticed that, for clarity, only the trajectories with a certain angle are shown. However, according to the conservation rules and simple geometric relations, the positions of focal points where the trajectories cross the $x$ axis do not change by varying the angle.

Figure 2

Spatial variations of (a) electron and (b) hole scaled excess densities defined by $\widetilde{\delta n}=\delta n/\left(V_0{n}_0^2\right)$. The electron injector is located at $x_0=-2.7W$ inside the first N lead, measured from the first interface. The density profiles are plotted inside the second normal contact ${\mathrm{N}}_2$. The superconducting layer has a width $W=0.5\xi$ and the potential barriers at the interfaces are $U_1=U_2=2\mu$ and other parameters are ${\mathrm{\Delta }}_0/\mu =0.01,\epsilon /{\mathrm{\Delta }}_0=0.8$. For both electron and hole sequences the focal points along the $x$ axis can be observed. The electron density shows an extra decaying push which indicates the EC contributions. The oscillations in the densities are due to the quantum interference of electronic waves upon scattering from the interfaces and the superconducting layer.

Figure 3

Local current densities of electrons $\left(J_e\right)$ and holes $\left(J_h\right)$ inside ${\mathrm{N}}_2$ along the $x$ axis and vs the distance from the second interface when (a) $U_1=U_2=\mu$, (b) $U_1=\mu ,U_2=0$, and (c) $U_1=0$ and $U_2=\mu$. (d)–(f) indicate the corresponding schematics in which the classical trajectories leading to the electron and hole focuses are shown. The electron and hole propagations are indicated by solid blue and dashed red lines, respectively. The width of the superconductor and injection point position are $W/\xi =0.22$ and $x_0/\xi =0.35$. The superconducting gap and excitation energy are assumed as ${\mathrm{\Delta }}_0/\mu =0.01$ and $\epsilon /{\mathrm{\Delta }}_0=0.8$. This figure clearly indicates that the sequence of focuses can be observed when both interfaces have a limited transparency $\left(U\ne 0\right)$. However, when the second or first interfaces are fully transparent $\left(U=0\right)$, only the first and second hole focuses survive, respectively.

Figure 4

Spatial variation of the local current densities for (a) electrons and (b) holes vs the distance from the second interface along the $x$ axis for various values of the superconducting layer width $W$. Although the EC signal indicated by electron current $J_e$ shows an overall decline for wider $W$, the CAR signal given by hole current $J_h$ is strong for $W/\xi \lesssim 1$. The parameters used for this plot have the values $x_0/\xi =0.4,U_1=U_2=\mu ,{\lambda }_F/\xi =0.04$, and $\epsilon /{\mathrm{\Delta }}_0=0.8$.

Figure 5

Spatial variation of (a) local electron current and (b) hole current along the $x$ axis for various distances of the injection point from the first interface $\left(x_0\right)$. The width of the superconductor is $W/\xi =0.5$ and other parameters are the same as those in Fig. 4.

Figure 6

Dependence of the hole and electron current densities at the first hole focus $\left(x_h^{\left(0\right)}\right)$ on the barrier strength $U$. Although the electron current decreases monotonically with $U$, the hole current at the focus becomes maximum at a certain strength of the potential barrier $U/\mu \sim 1$. All the parameters used here are the same as in Fig. 3.

Figure 7

Spatial variations of (a) electron and (b) hole densities inside the second normal contact ${\mathrm{N}}_2$ when there are two injection points separated by $\delta y=5{\lambda }_F$ in the $y$ direction. All parameters are the same as those used for Fig. 2. The corresponding focuses of each injection can be clearly seen beside the extra interferences of electron and hole waves induced by a two-point source.