Perfect crossed Andreev reflection in the proximitized graphene/superconductor/proximitized graphene junctions

We study the crossed Andreev reflection and the nonlocal transport in the proximitized graphene/ superconductor/proximitized graphene junctions with the pseudospin staggered potential and the intrinsic spin-orbit coupling. The crossed Andreev reflection with the local Andreev reflection and the elastic cotunneling being completely eliminated can be realized for the electrons with the specific spin-valley index when the intrinsic spin-orbit couplings in the left graphene and the right graphene possess the opposite sign. The perfect crossed Andreev reflection with its probability equal to $100\%$ can be obtained in the space consisting of the incident angle and the energy of the electrons. The crossed conductance and its oscillation with the superconductor length are also investigated. The energy ranges for the crossed Andreev reflection without the local Andreev reflection and the elastic cotunneling are clarified for the different magnitudes of the pseudospin potential and the spin-orbit coupling. The spin-valley index of the electrons responsible for the perfect crossed Andreev reflection can be switched by changing the sign of the intrinsic spin-orbit coupling or exchanging the biases applied on the left graphene and the right graphene. Our results are helpful for designing the flexible and high-efficiency Cooper pair splitter based on the spin-valley degree of freedom.

Figure 1

(a) The schematic illustration of the proximitized graphene/superconductor/proximitized graphene junctions. The dispersions for the $K\uparrow$ electrons (blue solid lines) and the $K^{\prime }\downarrow$ holes (blue dashed lines) in (b) the left graphene and (c) the right graphene. The dispersions for the $K\downarrow$ electrons (red solid lines) and the $K^{\prime }\uparrow$ holes (red dashed lines) in (d) the left graphene and (e) the right graphene. The red solid (hollow) circle denotes the $K\downarrow$ electron ($K^{\prime }\uparrow$ hole) and the black solid (dashed) arrows denote its motion direction. The scattering processes for the CAR without LAR and ECT of the $K\downarrow$ electrons are shown in (a), (d), and (e). The parameters $\mathrm{\Delta }=\lambda >0$ have been taken for the dispersions.

Figure 2

(a) The normal reflection probability and (b) the CAR probability of the incident $K\downarrow$ electrons for $L=0.55{\xi }_0$ in the $(E,\theta )$ space. The CAR probabilities of the incident $K\downarrow$ electrons for (c) $L=1.43{\xi }_0$ and (d) $L=2.53{\xi }_0$ in the $(E,\theta )$ space. The parameters are taken as ${\mathrm{\Delta }}_L={\mathrm{\Delta }}_R=0.5{\mathrm{\Delta }}_0$ and ${\lambda }_L=-{\lambda }_R=0.5{\mathrm{\Delta }}_0$.

Figure 3

(a) The crossed conductance spectra and (b) the oscillations of the crossed conductances with the length of superconductor for the injection of the $K\downarrow$ electrons. Other parameters are taken as the same as those in Fig. 2.

Figure 4

(a) The energy ranges for the CAR without LAR and ECT for the injection of the $K\downarrow$ electrons. The solid lines are the lower bounds while the dashed lines are the upper bounds for the different values of $\mathrm{\Delta }$. (b) The three zones surrounding the energy range for the CAR without LAR and ECT with $\mathrm{\Delta }=0.5{\mathrm{\Delta }}_0$.