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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., Supplementary information can be found at https://doi.org/10.1038/s42005-018-0097-1

We observe a series of sharp resonant features in the differential conductance of graphene-hexagonal boron nitride-graphene tunnel transistors over a wide range of bias voltages between 10 and 200 mV. We attribute them to electron tunneling assisted by the emission of phonons of well-defined energy. The bias voltages at which they occur are insensitive to the applied gate voltage and hence independent of the carrier densities in the graphene electrodes, so plasmonic effects can be ruled out. The phonon energies corresponding to the resonances are compared with the lattice dispersion curves of graphene-boron nitride heterostructures and are close to peaks in the single phonon density of states.