Fig. 1: Tunnel current through a localised state in a graphene-hBN tunnel transistor.

a Schematic band diagram of the device showing the energy level E₁ of the localised state in the barrier. Here, V_g and V_b are the applied gate and bias voltages, μ_B and μ_T are the chemical potentials on the bottom and top graphene layers measured with respect to the Dirac point on each layer, b is the tunnel barrier thickness and E_D is the energy difference between the Dirac points of each layer. b Measured steps in current-voltage, I(V_b), curves at V_g = 0 (red) and V_g = 1.65 V at a temperature T = 1.75 K and in zero magnetic field, B = 0, due to tunnelling through a single defect state. To compare the relative magnitude of the tunnelling current through one defect, the dashed black curve shows I(V_b) for a similar defect-free tunnel device. c I(V_b) curves at finite magnetic field, B = 3.75 T, T = 1.75 K and V_g = 0 (red) and V_g = 1.65 V (blue): the inset shows schematically the current flow when V_b = 0 and V_g > 0.