  Hexagonal boron nitride (hBN) is a large band gap layered crystal, frequently
incorporated in van der Waals (vdW) heterostructures as an insulating or tunnel
barrier. Localised states with energies within its band gap can emit visible
light, relevant to applications in nanophotonics and quantum information
processing. However, they also give rise to conducting channels, which can
induce electrical breakdown when a large voltage is applied. Here we use gated
tunnel transistors to study resonant electron tunnelling through the localised
states in few atomic-layer hBN barriers sandwiched between two monolayer
graphene electrodes. The measurements are used to determine the energy,
linewidth, tunnelling transmission probability, and depth within the barrier of
more than 50 distinct localised states. A three-step process of electron
percolation through two spatially separated localised states is also
investigated.
