Valley splitter and transverse valley focusing in twisted bilayer graphene

Abstract: 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. We show that such a valley splitter is obtained when the barrier varies slowly on the moir\'e 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.