Fig. 1: Inelastic resonant tunnelling through two adjacent localised states in hBN barrier, conceptualization.

a Schematic band diagrams illustrating elastic (top) and inelastic (bottom) resonant tunnelling events through localised states \(A\) and \(B\). b Tunnelling current (\(J\)) and differential tunnelling conductance curves at \(T=\) 2 \(\,{\rm{K}}\) as a function of bias (\({V}_{{\rm{b}}}\)) voltage at fixed backgate and topgate voltages of \({V}_{{\rm{bg}}}=\) 0 \(\,{\rm{V}}\) and \({V}_{{\rm{tg}}}=\) 2 \(\,{\rm{V}}\). The tunnel current and conductance values are normalized over the cross-sectional area of the Device 1 (\(S\approx\) 20 \({\mu }{{\rm{m}}}^{2}\)). The peak (step) in the conductance (current) bias voltage dependence, which is corresponding to inelastic tunnelling through states \(A\) and \(B\) in series (marked as \(A\)-> \(B\)) is about an order of magnitude greater than the one corresponding to elastic tunnelling through state \(A\) (marked as \(A\)). c Evaluation of inelastic tunnelling current through \(A\) to \(B\) for two coupling constants \({{|g|}}^{2}\), and elastic current for separate events of tunnelling through states \(A\) and \(B\) vs the transparency of the barrier, where \({\kappa }^{-1}\) is the length of the localization.