Tunable nonlocal valley-entangled Cooper pair splitter realized in bilayer-graphene van der Waals spin valves

Abstract

We investigate the Cooper pair splitting based on crossed Andreev reflection in a two-dimensional narrow superconductor coupled to two leads, where the two leads are van der Waals (vdW) spin valves formed by inserting bilayer graphene (BLG) between two antiparallel insulating ferromagnetic layers. In this BLG-based n-type vdW spin valve/superconductor/p-type vdW spin valve junction with edge-contact geometry, when controlling the Fermi energy and the interlayer electric field in BLG independently by top- and bottom-gate voltages in the vdW spin valve region, we first demonstrate that the two electrons of perfect Cooper pair splitting can remain nonlocal valley entanglement. This effect can be explained as follows: When both the two antiparallel insulating ferromagnetic layers and the interlayer electric field are taken into account, BLG exhibits a spin-splitting but valley-degenerate band gap near the Dirac point. Thus, our setups allow the Cooper pair in the BCS scenario to serve as a natural source of valley entanglement. We further predict large oscillations of cross conductance as a function of the doping of the superconductor, the junction length, and the bias voltage. Our results may provide a tool for realizing fast nonlocal switching of the valley-entanglement state.

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