Valley-polarised tunnelling currents in bilayer graphene tunnelling transistors

We study theoretically the electron current across a monolayer graphene/hexagonal boron nitride/bilayer graphene tunnelling junction in an external magnetic field perpendicular to the layers. We show that change in effective tunnelling barrier width for electrons on different graphene layers of bilayer graphene, coupled with the fact that its Landau level wave functions are not equally distributed amongst the layers with a distribution that is reversed between the two valleys, lead to valley polarisation of the tunnelling current. We estimate that valley polarisation $\sim 70\%$ can be achieved in high quality devices at $B=1$ T. Moreover, we demonstrate that strong valley polarisation can be obtained both in the limit of strong-momentum conserving tunnelling and in lower quality devices where this constraint is lifted.