We study transport in twisted bilayer graphene and show that electrostatic barriers can act as valley splitters, where electrons from the $K$ ($K$) valley are transmitted only to e.g. the top (bottom) layer, leading to valley-layer locked currents. W
e show that such a valley splitter is obtained when the barrier varies slowly on the moire scale and induces a Lifshitz transition across the junction, i.e. a change in the Fermi surface topology. Furthermore, we show that for a given valley the reflected and transmitted current are transversely deflected, as time-reversal symmetry is effectively broken in each valley separately, resulting in valley-selective transverse focusing at zero magnetic field.
Bilayer graphene hosts valley-chiral one dimensional modes at domain walls between regions of different interlayer potential or stacking order. When such a channel is brought into proximity to a superconductor, the two electrons of a Cooper pair whic
h tunnel into it move in opposite directions because they belong to different valleys related by the time-reversal symmetry. This is a kinetic variant of Cooper pair splitting, which requires neither Coulomb repulsion nor energy filtering but is enforced by the robustness of the valley isospin in the absence of atomic-scale defects. We derive an effective model for the guided modes in proximity to an s-wave superconductor, calculate the conductance carried by split and spin-entangled electron pairs, and interpret it as a result of local Andreev reflection processes, whereas crossed Andreev reflection is absent.
